UvA-DARE (Digital Academic Repository) Cyclic theory of ... · 508195-L-bw-Blom CYCLIC THEORY OF LIE ALGEBROIDS ... Prof. dr. S. V. Shadrin ... Allereerst wil ik Hessel Posthuma bijzonder

UvA-DARE is a service provided by the library of the University of Amsterdam (http://dare.uva.nl)

UvA-DARE (Digital Academic Repository)

Cyclic theory of Lie algebroids

Blom, A.

Link to publication

Citation for published version (APA):Blom, A. (2017). Cyclic theory of Lie algebroids

General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s),other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).

Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, statingyour reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Askthe Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam,The Netherlands. You will be contacted as soon as possible.

Download date: 20 Jul 2018

http://dare.uva.nl/personal/pure/en/publications/cyclic-theory-of-lie-algebroids(a6879f65-d429-4ebb-88d9-9499a0bb43e1).html

508195-L-os-Blom508195-L-os-Blom508195-L-os-Blom508195-L-os-Blom Processed on: 20-3-2017Processed on: 20-3-2017Processed on: 20-3-2017Processed on: 20-3-2017

Cyclic theory of Lie algebroids Arie Blom


Arie B

lom

Boekenlegger nog aan te leveren door klant

508195-os-Blom.indd 1,6 10-02-17 11:25

Processed on: 20-2-2017Processed on: 20-2-2017Processed on: 20-2-2017Processed on: 20-2-2017

508195-L-bw-Blom508195-L-bw-Blom508195-L-bw-Blom508195-L-bw-Blom

Cover design by Arie Blom.

Typeset by LATEX.

ISBN: 978-94-028-0527-7

Publisher: Ipskamp Printing

NUR code: 921 (Fundamentele wiskunde)



CYCLIC THEORY OF LIE ALGEBROIDS

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

prof. dr. ir. K.I.J. Maex

ten overstaan van een door het College voor Promoties

ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op donderdag 2 maart 2017, te 12:00 uur

door

Arie Blomgeboren te Breda



Promotiecommissie

Promotor: Prof. dr. E. M. Opdam (Universiteit van Amsterdam)

Copromotor: Dr. H. B. Posthuma (Universiteit van Amsterdam)

Overige leden: Dr. R. R. J. Bocklandt (Universiteit van Amsterdam)

Prof. dr. R. Nest (University of Copenhagen)

Prof. dr. S. V. Shadrin (Universiteit van Amsterdam)

Prof. dr. L. D. J. Taelman (Universiteit van Amsterdam)

Porf. dr. X. Tang (Washington University)

Prof. dr. P. Xu (Pennsylvania State University)

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

The research for this thesis was supported by NWO through the Geometry and Quantum

Theory cluster under grant nr. 612.009.013.





Contents

Dankwoord 1

Introduction 3

1 Preliminaries 11

1.1 Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Three types of Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.2 Alternative definitions of Lie algebroids . . . . . . . . . . . . . . . . . . . 17

1.1.3 Lie algebroid cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 Noncommutative geometry of Lie algebroids . . . . . . . . . . . . . . . . . . . . . 21

1.2.1 The universal enveloping algebra of a Lie algebroid . . . . . . . . . . . . . 22

1.2.2 Hochschild and Cyclic theory . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 The Poincare–Birkhoff–Witt theorem 29

2.1 The universal enveloping algebra and the jet algebra . . . . . . . . . . . . . . . . 30

2.1.1 The universal enveloping algebra revisited . . . . . . . . . . . . . . . . . . 30

2.1.2 The jet algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Incarnations and properties of the PBW theorem . . . . . . . . . . . . . . . . . . 34

2.2.1 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.2 Compatibility with the smooth approach . . . . . . . . . . . . . . . . . . 38

2.2.3 The noncommutative PBW theorem . . . . . . . . . . . . . . . . . . . . . 40

2.2.4 The derivative of the PBW map . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.5 Twisting by a locally free sheaf . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Cyclic theory of the universal enveloping algebra 47

3.1 Poisson (co)homology and representations up to homotopy . . . . . . . . . . . . . 47

3.1.1 Polynomial polyvector fields and differential forms . . . . . . . . . . . . . 48

3.1.2 The Poisson structure on A∨ . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.3 Representations up to homotopy . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.4 Poisson (co)homology and the modular class . . . . . . . . . . . . . . . . 54

3.1.5 Poisson (co)homology vs. representations up to homotopy . . . . . . . . . 57

3.2 Application of formality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 Deformation quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.2 The basic complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65



CONTENTS

3.2.3 Computation of the cyclic theory . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 The trace density map 71

4.1 Definition and properties of the trace density map . . . . . . . . . . . . . . . . . 72

4.1.1 The set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.2 The Fedosov resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.3 A proof of Theorem 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 The index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.2 The index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.3 Compatibility with the HKR map . . . . . . . . . . . . . . . . . . . . . . 84

4.2.4 Holomorphic Lie algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A Formality 89

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A.2 Curved L∞-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

A.3 Polyvector fields and polydifferential operators . . . . . . . . . . . . . . . . . . . 93

A.4 Differential forms and Hochschild chains . . . . . . . . . . . . . . . . . . . . . . . 94

A.5 Formality for Hochschild L-(co)chains . . . . . . . . . . . . . . . . . . . . . . . . 95

A.6 Cyclic L-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

B The local Riemann–Roch theorem 103

B.1 The Weyl algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.2 The Hochschild cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.3 The map to Lie algebra cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.4 The fundamental cyclic cocycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.5 The local Riemann–Roch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 107

Samenvatting 111

Summary 112



Dankwoord

Allereerst wil ik Hessel Posthuma bijzonder veel bedanken voor al zijn hulp in de afgelopen

jaren. Elke week weer, of ik nu was opgeschoten of niet, of ik nuttige artikelen had gelezen

of niet, of ik een redelijk idee had gehad of niet, luisterde hij naar mijn pogingen om mezelf

wiskundig verstaanbaar te maken en zette hij zijn altijd weer uitwaaierende eigen ideeen uiteen.

Menigmaal bekeek hij een probleem waar ik al dagen niet uitkwam op een heel andere manier,

waardoor een oplossing ineens weer binnen bereik lag.

The members of the committee who took the effort to read an comment on the thesis,

which contained, to be honest, too many spelling and other mistakes at the time, deserve a

word of thanks. Morever, I want to thank Eric Opdam for his flexibility with regards to certain

deadlines when this was needed, especially because I informed him rather late about the matter.

Ik ben vrijwel altijd met plezier naar de universiteit gegaan, en dat is niet in de laatste

plaats te danken aan de leuke mensen met wie ik een kamer heb gedeeld in de afgelopen jaren.

Buiten mijn kamer, soms gebogen over een klein wiskundig probleem of discussierend over alles

wat ter sprake kwam, ben ik me samen met Dirk en Apo te buiten gegaan aan de heerlijke

koffie uit de koffiemachine, waarvoor ik de universiteit graag wil bedanken. Voordat ik aan

mijn promotie begon heb ik talloze uren met Simen (en anderen) sommetjes gemaakt, en ik

dank hem voor de precisie die ik soms miste.

Gedurende de tijd dat ik aan mijn proefschrift werkte, heb ik samengewoond met verschei-

dene (groepen) mensen, die, hoewel de gesprekken op wiskundig gebied meestal nogal moeizaam

verliepen, toch altijd een steun zijn geweest. Dank jullie voor de nodige afleiding! Nu ik erover

nadenk, wil ik ook graag de gemeente Amsterdam bedanken voor het feit dat ik op aangename

wijze in hun vastgoed heb kunnen wonen in deze tijd.

Mijn studenten waren -meestal toch, enthousiast en stelde velerlei vragen op verscheidene

manieren, wat me de stof van verschillende kanten deed bekijken, en dat heb ik zeer gewaardeerd.

Mijn familie tenslotte heeft, hoewel er wat betreft de wiskundige conversaties een soortgelijk

euvel bestond als bij de voorgenoemde huisgenoten, me altijd mijn gang laten gaan en gesteund

met liefde en vertrouwen, en daar ben ik ook dankbaar voor.

1



2



Introduction

Lie algebroids

The central objects of interest of this thesis are Lie algebroids. They play a prominent role

in geometry as generalized infinitesimal symmetries of spaces. In the smooth category, it was

Pradines who introduced the concept of a Lie algebroid in [Pr], and he studied its relation to the

corresponding global symmetries of spaces, called Lie groupoids. However, the core idea, the

study of vector fields preserving some geometry or PDE which are closed under the Lie bracket,

has a much longer history and was initiated in the works of Lie and Cartan. Lie algebroids can

be defined in different contexts, but they always play the role of generalized symmetries. In

this thesis, we consider:

1. The differential geometric version; smooth Lie algebroids.

2. The algebraic notion of Lie–Rinehart algebras

3. The sheaf-theoretic version; sheaves of Lie algebroids over a ringed space.

The notion of Lie–Rinehart algebras is introduced by a number of authors under different

names, see [M] for an extensive overview. Rinehart showed in [Rin] that the cohomology theory

of Lie–Rinehart algebras can be subsumed in the theory of homological algebra, and he proved

the analogue of the Poincare–Birkhoff–Witt theorem for the associated universal object of a

Lie–Rinehart algebra, which will be discussed later in this introduction. The first two sections of

[BB] treat Lie algebroids in algebraic geometry, which are examples of sheaves of Lie algebroids

over ringed spaces. To stay close to the geometric intuition, we mostly consider smooth Lie

algebroids in this introduction:

Definition. A smooth Lie algebroid is a vector bundle π : A → M, together with the following

data:

1. A smooth vector bundle map ρ : A → TM, called the anchor map of the Lie algebroid.

2. A Lie bracket on the space of smooth sections Γ(A) of A such that for all s, t ∈ A and

f ∈ C∞(M), the following Leibniz identity holds:

[s, ft]A = Lρ(s)(f)t+ f[s, t]A.

Here L denotes the Lie derivative of vector fields.

Perhaps the most well-known examples of smooth Lie algebroids are tangent bundles of

smooth manifolds (where the anchor map is just the identity) and finite dimensional, real Lie

algebras (where M is a point). Other examples of Lie algebroids are given by regular foliations,

which correspond (by the Frobenius theorem) to Lie algebroids with an injective anchor map,

actions of Lie algebras on a manifold M, and manifolds with a Poisson structure. Many of

the concepts in the theory around Lie algebroids are generalizations of concepts in either the

3



4

theory around tangent bundles or the theory around Lie algebras. For example, one has the

A-de Rham complex

Ω0A

dA−→ Ω1A

dA−→ Ω2A

dA−→ . . .

associated to a Lie algebroid A (so d2A = 0), which both generalizes the Chevalley–Eilenberg

complex of Lie algebras and the de Rham complex of smooth manifolds. The cohomology of

this complex, denoted by H•(A), is called Lie algebroid cohomology and reduces to de Rham

cohomology and Lie algebra cohomology in the aforementioned examples. Dual to the A-de

Rham forms, one has the A-polyvector fields TApoly with a graded Lie bracket [ , ]SN -called

the Schouten–Nijenhuis bracket, obtained as the natural extension of the Lie bracket on A.

This generalizes the multivector vector fields Tpoly(M) on a smooth manifold, and in fact all

the structures which comprise the Cartan calculus on the pair (Tpoly(M),Ω(M)): the wedge

products on Tpoly(M) and Ω(M), the insertion operator ι for polyvector fields in differential

forms, the bracket [ , ]SN on Tpoly(M), the de Rham differential d and the Lie derivative

L = dι ± ιd, can be defined in the same way for the pair (TApoly,ΩA). We refer to §1.1 for the

precise definitions and a discussion of these constructions.

The universal enveloping algebra U(A) of a Lie algebroid A is the algebra generated by Γ(A)

and C∞(M), subject to relations

[s, t]A = st− ts, ρ(s)(f) = [s, f], f · s = fs.

Consult §1.2 for a precise definition. It generalizes the universal enveloping algebra U(g) as-

sociated to a Lie algebra g. For A = TM, the universal enveloping algebra is given by the

differential operators Diff(M) on M. Like the universal enveloping algebra of a Lie algebra,

it comes equipped with a natural ascending filtration. It is clear from this definition that the

universal enveloping algebra is noncommutative when the Lie bracket and the anchor map of A

are non trivial. When A is a Lie algebroid with trivial bracket and anchor, i.e. a vector bundle,

one has U(A) = SymA; the symmetric algebra bundle of A.

Finite dimensional Lie algebras arise as infinitesimal objects of Lie groups; more precisely as

the collection of right (or left) invariant vector fields with respect to the right (or left) action of

the group on itself. The notion of Lie groupoids, which generalizes smooth Lie groups, provides

for an analogon of this construction, in the sense that they are smooth objects together with

several operations to which one can associate infinitesimal, invariant vector fields which form

smooth Lie algebroids. One can thus associate a Lie algebroid to each Lie groupoid, The

converse statement however, which is Lie’s third theorem in the case of Lie groups, does not

hold. The precise obstructions to “integrability” are given in [CF1].

Cyclic theory

Given a unitary, associative algebra C, one can define its Hochschild and cyclic homology

as well as their dual cohomology theories, commonly referred to as the ”cyclic theory of an

algebra”. They are a fundamental ingredient in noncommutative geometry. Their definition is

given in terms of explicit complexes, see §1.2.2, equipped with two differentials; the Hochschild

differential b and the cyclic differential B. The latter was already contained in the work of

Rinehart, c.f. [Rin], but lay dormant until it was rediscovered in the works of Connes [Co]

and Feigin and Tsygan [FT]. The importance of cyclic theory is explained by considering the

commutative case:

For a smooth commutative algebra C, the Hochschild homology H•(C) of C is isomorphic to

the algebraic de Rham forms [Rin], courtesy of the Hochschild–Konstant–Rosenberg theorem of

[HKR], and the cyclic differential B corresponds to the de Rham differential d. The Hochschild

cohomology H•(C,C) of C is isomorphic to the algebraic polyvector fields on C, and one can



INTRODUCTION 5

define operations which give the pair (H•(C,C), H•(C)) the structure of a Cartan calculus,

see [DTTs]. These operations correspond to the Cartan calculus structure on the algebraic

polyvector fields and differential forms. When C = C∞(M), one has to take into account the

locally convex topology on the algebra C∞(M) in the definition of the cyclic theory, and Connes

proved in [Co], that

(H•(C,C), H•(C)) ∼= (Tpoly(M),Ω(M)).

The virtue of cyclic theory is that it is naturally defined for noncommutative algebras C, lead-

ing to ”de Rham cohomology for noncommutative algebras”. This can be applied to certain

pathalogical spaces which do not have a smooth structure, but to which one can associate a

natural noncommutative algebra. The cyclic theory of this algebra is a replacement for the

usual Cartan calculus for smooth spaces. One of the earliest applications of cyclic theory is

index theory for foliations, in which the cyclic homology is the recipient of the noncommutative

Chern character, c.f. [Co].

In general, the cyclic theory of a noncommutative algebra is quite hard to compute, but in

certain cases the result is known. In [Wo], Wodzicki computed the Hochschild and cyclic homol-

ogy of the algebra of differential operators on a manifold, and Kassel [Ka] did the same for the

universal enveloping algebra of a Lie algebra. Both of these examples are almost commutative,

i.e. the algebras admit a filtration such that the associated graded algebras are commutative,

and the associated graded algebras inherit a Poisson structure measuring the non commutativ-

ity of the original algebra. In the first case, this Poisson structure corresponds to the well-known

symplectic structure on T∗M, and in the latter case it is given by the linear Poisson structure

on g∗. In both cases the cyclic theory of the noncommutative algebra is computed in terms

of Poisson (co)homology. A great part of this thesis is devoted to the cyclic theory of the

universal enveloping algebra of a Lie algebroid, of which the two previous cases are examples.

The Poisson structure on A∨ plays a similar role in the computation of the cyclic theory of the

universal enveloping algebra.

The Poincare–Birkhoff–Witt theorem

The Poincare–Birkhoff–Witt theorem for Lie algebras states that gr(U(g)) ∼= Sym(g) for a Lie

algebra g over a field K, where Sym(g) is the symmetric algebra of g. If Q ⊂ K, one can define

the symmetrization map Sym(g) → U(g) inducing the isomorphism on the associated graded

spaces. In [Rin], Rinehart proved in a more algebraic setting that the associated graded object

to the universal enveloping algebra of a projective Lie–Rinehart algebra L is isomorphic to the

symmetric algebra generated by L, which implies that gr(U(A)) ∼= Sym(A) for a smooth Lie

algebroidA. However, he did not construct a map Sym(A) −→ U(A) inducing this isomorphism.

In [NWX] the concept of a local integrating groupoid was used define such a map in the smooth

case.

We generalize the map of [NWX] to locally free sheaves of Lie algebroids L of constant

rank r over a locally ringed space (X,OX) over a field K ⊃ Q, provided they are equipped with a

global L-connection ∇L on L. This condition is certainly satisfied for all smooth Lie algebroids.

Roughly speaking, we replace the local integrating groupoid of [NWX] by the sheaf of L-Jets

over X

J(L) := HomOX(U(L),OX),

Using the L-connection, we define an exponential map which we use to prove the dual version

of the PBW theorem:

Theorem. Let L be a locally free sheaf of Lie algebroids of constant rank r together with an

L-connection ∇L, then there exists a map

j∇ : J(L) −→ SymOX(L∨),



6

which is an OX-linear isomorphism of sheaves of algebras with respect to the first OX-module

structure on J(L).

Dualizing this result gives an isomorphism Sym(L) → U(L) which respects the coalgebra

structures. We give a recursive relation that defines this map, and this relation turns out to

be the same as the one in [LSX14]. In the applications of the PBW theorem that we consider;

the construction of a trace density map and a formality result, we use the version of the PBW

theorem as stated in the theorem, so in our view this is the more fundamental result. Also, the

definition of the morphism is natural and easy, which makes the study of the properties of the

PBW map more accessible.

Cyclic theory of the universal enveloping algebra

Extending the results of Kassel and Wodzicki, we compute the cyclic theory of the universal

enveloping algebra of a smooth Lie algebroid A → M. First we introduce the structures that

are needed to phrase the main results.

Representations up to homotopy

Representations of Lie algebroids A → M are given by flat Lie algebroid connections on vector

bundles E → M. In this sense, Lie algebroids do not have many representations. For example,

there is no adjoint representation in general. A more flexible notion of a representation up

to homotopy was defined in [AC]. Informally speaking, the vector bundle E is replaced by a

complex of vector bundles . . .∂→ Ei ∂→ Ei+1 ∂→ . . ., and the flat A connection on E is replaced

by not necessarily flat connections on each bundle Ei which are compatible with the maps ∂, i.e.

∇∂−∂∇ = 0. The data is completed by a collection of tensors satisfying relations which imply

that, together with ∂ and ∇, they form a derivation D of degree 1 which squares to zero on the

total complex ΩA(⊕

i Ei). Such differentials are also known as flat superconnections on a Z-

graded bundle and they were introduced in the Z/2-graded case by Quillen in [Qu]. One of the

virtues of this definition is that for any smooth Lie algebroid A, the adjoint representation of A,

denoted by adA, can be defined. Although it depends on the choice of a connection on A, the

associated cohomology is canonical. Moreover, the category of representations up to homotopy

is symmetric and monoidal, hence one can define Sym(adA), the symmetric algebra of the

adjoint representation of A. One of the main results of this thesis computes the Hochschild

cohomology and homology of the universal enveloping algebra.

Theorem. Let A be a smooth Lie algebroid of rank r over a smooth manifold of dimension n

with universal enveloping algebra U(A).

i) For the Hochschild cohomology, there is a natural isomorphism

H•(U(A),U(A)) ∼= H•Lie((Sym(adA)).

ii) When both M and A are orientable, there is a natural isomorphism

H•(U(A)) ∼= Hn+r−•Lie (Sym(adA)⊗QA)

for the Hochschild homology.

The bundle QA is a natural representation of A, sometimes referred to as the bundle of

transverse volume forms. In the special cases A = TM and A = g we recover exactly the results

of Wodzicki and Kassel, c.f. [Ka, Wo]. Let us give an informal discussion of the proof. For

simplicity we restrict ourselves to part i) of the theorem.

Discussion of the proof: step 1

The PBW theorem induces a natural Poisson structure on the associated graded algebra of



INTRODUCTION 7

U(A), and the differential of the first page of the spectral sequence that can be associated to

the filtration on the Hochschild cocomplex of U(A) turns out to be the Poisson cohomology

differential associated with this Poisson structure, c.f. [Br].

Our first step is to identify the Poisson cohomology complex associated to the linear Poisson

structure on the dual of A with the complex (ΩA(Sym(adA), D). In the second step we

construct a quasi-isomorphism between the Hochschild cohomology complex for U(A) and the

Poisson cohomology complex of the aforementioned Poisson structure.

Let us first discuss the first step. We start with the observation that sections of SymA

correspond to functions on A∨ which are polynomial along the fibers. The Lie bracket and the

anchor map ρ define a Poisson structure on SymA in a natural way, and this can be extended

to all smooth functions on A∨. The derivations L := DerR(SymA), which are a model for the

vector fields on A∨ which are polynomial along the fibers of the projection A∨ → M, form a

sheaf of Lie algebroids over the locally ringed space (M, SymA), and it fits into the following

short exact sequence of sheaves of Lie algebroids over (M, SymA):

0 −→ SymA⊗A∨ −→ DerR(SymA) −→ SymA⊗ TM −→ 0.

The choice of a connection on A splits this sequence, and such a splitting defines a Lie bracket

on SymA ⊗ (A∨ ⊕ TM). One can do the same for the sheaves of polyvector fields TLpoly(A

∨)

and differential forms ΩL(A∨) associated to L. These are models for the polyvector fields and

differential forms on A∨ which are polynomial along the fibers. The linear Poisson structure

on SymA corresponds to a Maurer–Cartan element θ ∈ TLpoly(A

∨), and we show that under

the identification induced by the connection:

• The Poisson cohomology differential [θ, ]SN corresponds to the differential D of the com-

plex Ω•(Sym(adA)) of the symmetric algebra of the adjoint representation.

• The Poisson homology differential Lθ corresponds to the differential D of the complex

Ω•(Sym(adA)⊗QA)

The linear part of the identification between the Poisson cohomology complex and the adjoint

representation of A was already contained in [CM].

Step 2: formality

The second step relies on the formality theorem for Lie algebroids. Let us first bring the

original formality theorem that Kontsevich proved in [K] to the attention of the reader. Given

a smooth manifold, M one can construct the complex of polydifferential operators Dpoly(M),

which admits a graded Lie algebra bracket [ , ]G and a Hochschild differential [m, ] = bH.

Hence, it is a differential graded Lie algebra (dg Lie algebra). It forms a local model for the

Hochschild cohomology of C∞(M). By the smooth HKR theorem, the symmetrization map

HKR : Tpoly(M) −→ Dpoly(M)

defines an isomorphism on the cohomology, however it does not respect the Lie algebra struc-

tures [ , ]SN and [ , ]G. The formality theorem states that this defect can be cured if one

considers the more flexible notion of L∞-morphisms; i.e. Kontsevich proved the existence of an

L∞-quasi-isomorphism

K : Tpoly(M) −→ Dpoly(M).

One of the main features of L∞-morphisms is that they relate Maurer–Cartan elements. If one

introduces a formal variable h on both sides, Maurer–Cartan elements in the LHS are given by

(formal) Poisson structures on M, and on the RHS they define formal deformations of C∞(M),

i.e. -products on C∞(M)[[h]]. Given a Maurer–Cartan element π ∈ Tpoly(M), one can twist

the L∞-morphism K by this element, which gives a morphism of complexes:

Kπ : (Tpoly(M)[[h]], [π, ]SN) −→(Dpoly(M)[[h]], bH + [π, ]G

).



8

Moreover, there exists a natural map from the complex on the RHS to the Hochschild coho-

mology complex of the deformed algebra (C∞(M)[[h]], ).

In the appendix we revisit the formality theorem for a certain type of sheaves of Lie alge-

broids, which encompasses the sheaf of Lie algebroids L = DerR(SymA) over (M, SymA). The

proof that we give stipulates the role of the Poincare–Birkhoff–Witt theorem for Lie algebroids.

The result is an L∞-quasi-isomorphism of dg Lie algebras:

U : TLpoly(A

∨)−→DLpoly(A

∨)

with the RHS given by the L-polydifferential operators on SymA. We twist this L∞-morphism

by the Maurer–Cartan element θ:

Uθ :(TLpoly(A

∨), [θ, ]SN)−→

(DL

poly(A∨), bH + [θ, ]G

).

Then we show that the deformation of SymA which is associated to the element θ ∈ DLpoly(A

∨)

is isomorphic to U(Ah), where Ah is the adiabatic Lie algebroid associated to A c.f. [NWX].

In particular, setting h = 1, we obtain a natural morphism:

(DL

poly(A∨)[[h]], bH + [θ, ]G

)−→

(C•(U(A),U(A)), bH

).

Finally, the formality map composed with this morphism gives a map

(TLpoly(A

∨), [θ, ]SN)−→ C•(U(A),U(A), bH).

This map respects the natural filtrations on both complexes and using a spectral sequence

argument we show that the morphisms are in fact quasi-isomorphisms, which implies that

H•θ(A

∨) ∼= H•(U(A),U(A)),

where the LHS is the cohomology of the complex(TLpoly(A

∨), [θ, ]SN

), i.e. the Polynomial

Poisson cohomology of A∨. Combining this result with step 1) gives part i) of the theorem.

The dual version of the formality theorem which is, in the local case, proved by Shoikhet in [S],

leads to a proof of part ii).

The trace density map

The notion of a trace density map comes from deformation quantization and the algebraic index

theorem proved by Nest and Tsygan in [NeTs95]. The same authors described the relation

between the algebraic index theorem and the analytic index theorem in [NeTs96]. We refer

to the introduction of chapter 4 for a longer discussion. From now on, we assume that L is

either the sheaf of sections of a holomorphic Lie algebroid or a smooth Lie algebroid. The main

difference is that the former does, in general, not admit global L-connections on L. To phrase

the main theorem of the last chapter, we use the language of derived categories:

Theorem. Let L be a either a smooth or a holomorphic Lie algebroid and E a locally free

L-module.

i) In the derived category D(X) of K-sheaves on X there exists a canonical morphism

Φ ∈ HomD(X)

((CC•(U(L)) , b+ u−1B

),(TotΩ2r−•

L , dL

)).

ii) Restricted to the structure sheaf OX ⊂ U(L,E), the following diagram commutes:

(CC•(OX), b+ u−1B

) (Ω•

L, u−1dL

)

(CC•(U(L,E)), b+ u−1B

) (Ω2r−•

L , dL

).

i

HKRuL

∪TdLChL(E)

Φ



INTRODUCTION 9

Let us clarify this theorem. When L = A is smooth, part i) implies that there exists a

canonical morphism

Φ : HP•(U(A)) −→ HLie(A)

from the periodic cyclic homology (part of the ”cyclic theory”) of U(A) to the Lie algebroid

cohomology of A.

When L = A is the sheaf of sections of a holomorphic Lie algebroid, part i) of the theorem

implies that there is a canonical morphism between the hyper(co)homology

H•(CC•(U(A)), b+ u−1B) −→ H2r−•(ΩA, dA).

The RHS is the Lie algebroid cohomology of A, and the LHS is the periodic cyclic homology

of the sheaf U(A). In the holomorphic case part ii) of the theorem measures the failure of the

trace density map to be compatible with the HKRA map in terms of Lie algebroid characteristic

classes. The hyper(co)homology of both sides can be computed by Dolbeault-type resolutions,

defined in [LSX12].

Outline of the chapters

Let us give a brief description of what is done in each chapter. We start each chapter with a

more extensive description of its contents.

Chapter 1 In the first chapter we define Lie algebroids from three different points of view;

smooth Lie algebroids, Lie–Rinehart algebras and sheaves of Lie algebroids. Moreover, we give

three equivalent definitions of the structure of a Lie algebroid, after which we briefly discuss

Lie algebroid cohomology and give some examples.. We finish with the definitions of the cyclic

theory of an algebra.

Chapter 2 The second chapter is devoted to the PBW theorem for Lie algebroids. We start

with a discussion of the various structures on the universal enveloping algebra and the jet algebra

associated to a Lie algebroid. Then we prove the PBW theorem. Finally, we discuss the time-

dependence of the PBW theorem given a linear combination of connections and incorporate the

so-called twisting by a locally free OX-module E.

Chapter 3 In this chapter we compute the cyclic theory of the universal enveloping algebra

of a smooth Lie algebroid. It starts with identification of the Poisson (co)homology complexes

for the linear Poisson structure on A∨ with the complex of the symmetric algebra of the

adjoint representation of A (twisted by the bundle QA). Then we discuss the application of

the formality map to prove the main results.

Chapter 4 We discuss the construction of the trace density morphism for Lie algebroids. It

starts with the notion of a Fedosov resolution and the application of the PBW theorem. Since

global connections are not necessarily given, we have to use Cech resolution of complexes, hence

the trace density morphism is defined in the derived category. To prove that it commutes with

the differentials we study the time dependence of the PBW theorem in more detail. Finally, we

introduce characteristic classes and compute the class of 1 under the trace density map, and

discuss the compatibility with the HKR map.

Chapters 2 and 4 are based on the preprint [BP] and chapter 3 is based on a forthcoming

paper.



10



Chapter 1

Preliminaries

1.1 Lie algebroids

The central objects of interest of this thesis are Lie algebroids. In general, Lie algebroids can

be defined as sheaves of Lie algebroids over a ringed space, encompassing for example smooth

Lie algebroids, holomorphic Lie algebroids and algebraic Lie algebroids. We will also define the

notion of Lie–Rinehart algebras, which are the algebraic counterparts of Lie algebroids, and

we will discuss the various relations between smooth Lie algebroids, Lie–Rinehart algebras and

sheaves of Lie algebroids.

Perhaps the most well-known examples of smooth Lie algebroids are tangent bundles of

smooth manifolds and finite dimensional, real Lie algebras. The theory around smooth Lie

algebroids, and more generally sheaves of Lie algebroids, is for a great deal analogous to the

theory of tangent bundles and the theory of Lie algebras. For example, one has the de Rham

complex associated to a Lie algebroid, which both generalizes the Chevalley–Eilenberg complex

of Lie algebras and the de Rham complex of smooth manifolds, and thus Lie algebra cohomol-

ogy and de Rham cohomology are special instances of Lie algebroid cohomology. Moreover,

analogous to the dg Lie algebra of multivector vector fields of a smooth manifold, one can

define the dg Lie algebra of A-polyvector fields for a given Lie algebroid A. After reviewing

these constructions, we will give three equivalent definitions of the structure of a sheaf of Lie

algebroids on a given OX-module L, where (X,OX) is a ringed space.

Finite dimensional Lie algebras arise as infinitesimal objects of Lie groups; more precisely as

the collection of right (or left) invariant vector fields with respect to the right (or left) action of

the group on itself. The notion of Lie groupoids, which generalizes smooth Lie groups, provides

for an analogon of this construction, in the sense that they are smooth objects together with

several operations to which one can associate infinitesimal, invariant vector fields which form

smooth Lie algebroids. Lie’s third theorem however, that gives a unique correspondence between

real, finite dimensional Lie algebras and simply connected Lie groups, cannot be extended to

the realm of Lie groupoids and Lie algebroids, because not all smooth Lie algebroids admit an

integration.

1.1.1 Three types of Lie algebroids

In this thesis we discuss three types of Lie algebroids: the differential geometrical variant,

which are given by smooth Lie algebroids; the algebraic notion of Lie–Rinehart algebras and

the sheaf-theoretic version, provided by sheaves of Lie algebroids over ringed spaces. We start

with the smooth case. Let M be a smooth manifold.

11



12 CHAPTER 1. PRELIMINARIES

Definition 1.1.1. A smooth Lie algebroid is a vector bundle π : A → M together with the

following data:

• A smooth vector bundle map ρ : A → TM.

• A Lie bracket on the space of smooth sections Γ(A) of A such that for all α,β ∈ A and

f ∈ C∞(M), the follolwing Leibniz identity holds:

[α, fβ]A = Lρ(α)(f)β+ f[α,β]A.

where L denotes the usual Lie derivative of vector fields.

Remark 1.1.2.

1) A computation in [Mar], involving the Jacobi identity for the bracket [ , ]A, shows that ρ

induces a Lie algebra morphism ρ : Γ(A) → Γ(TM) on spaces of sections, where the RHS is

endowed with the well-known Lie bracket of vector fields.

2) A computation involving the Leibniz identity, for which we again refer to [Mar] shows that

the Lie bracket is local ; this means that the value α,β(x) for two sections α,β ∈ Γ(A) at a

point x ∈ M is determined by the first order jets of α and β. In partcicular, this means that

the bracket can equivalently defined as a bracket on the sheaf of sections of A.

Example 1. The tangent bundle of M with the Lie bracket given by the commutator of vector

fields, and the anchor map equal to the identity is a smooth Lie algebroid.

Example 2. The same holds for a finite dimension real Lie algebra g over a point pt with

the Lie bracket of g and the trivial anchor map.

The algebraic counterpart of Lie algebroids are given by Lie–Rinehart algebras, [Rin]. Their

definition reads as follows:

Definition 1.1.3. Given a commutative ring K, a Lie–Rinehart algebra (L, R) is given by a

commutative K-algebra R and a Lie algebra (L, [ , ]) over K, which is a left R-module and which

acts on R by derivations. Denoting the Lie bracket by [ , ], the module structure by r⊗ l → rl,

and the action by l⊗ r → l(r) for r ∈ R and l ∈ L, the structures are compatible in the following

way:

(r1l)(r2) =r1(l1(r2)),

[l1, rl2] =l1(r)l2 + r[l1, l2].

Remark 1.1.4. In this thesis we assume that R is commutative and unital, K is either R or

C and L is projective over R. Many of the constructions for Lie–Rinehart algebras can be done

in greater generality, i.e. when K is replaced by a commutative, unital ring k. The condition

that L is projective implies the PBW theorem of [Rin], and, under the extra condition that

L has constant, finite rank, is used in [Hue] and references therein to prove duality results for

(co)homology. Lie–Rinehart algebras are also called Lie–Rinehart pairs.

Example 3. A standard example of a Lie–Rinehart algebra is given by a commutative ring

R and the Lie algebra of derivations Derk(R) of R with the commutator bracket. This is the

algebraic counterpart of the tangent bundle example.

Example 4. The global sections L = Γ(A) of a smooth Lie algebra A → M and the algebra

of smooth functions R = C∞(M) form a Lie–Rinehart algebra (Γ(A), C∞(M)). Conversely, due

to the Serre–Swan theorem, any finitely generated, projective module L over C∞(M) such that

(L,C∞(M)) is a Lie–Rinehart algebra, is given by the sections of a smooth Lie algebroid.



1.1. LIE ALGEBROIDS 13

The most general form of Lie algebroids that we will study consists of sheaves of Lie al-

gebroids over a given (locally) ringed space (X,OX) such that OX is a sheaf of K-algebras. A

detailed introduction into Lie algebroids in algebraic geometry, which are examples of sheaves

of Lie algebroids over ringed spaces, can be found in the first two sections of [BB].

Definition 1.1.5. A Lie algebroid L over (X,OX) is both a sheaf of OX-modules and a sheaf

of K-Lie algebras over X, which acts by derivations on OX. The compatibilities between these

structures are given by

f1(l1(f2)) = f1l1(f2)

[l1, f1l2] = l1(f1)l2 + f1[l1, l2]

for all f1, f2 ∈ OX and l1, l2 ∈ L, where we used similar notations as in Definition 1.1.3.

Remark 1.1.6. We assume that L is locally free and of finite, constant rank over OX. There

are interesting examples of sheaves of Lie algebroids that do not have this property, e.g. in [P]

Lie algebroids are defined as coherent sheaves over analytic spaces.

Let us now give some examples and relations between smooth Lie algebroids, Lie–Rinehart

algebras and sheaves of Lie algebroids.

Examples

1. Any smooth Lie algebroid A → M gives rise to the Lie–Rinehart algebra (Γ(A), C∞(M))

given by the global sections and to a sheaf of Lie algebroids ΓA over the locally ringed

space (C∞M,M), where C∞

M is the sheaf of smooth functions on M.

2. Given a Lie–Rinehart algebra (L, R), we denote by (Spec(R),OSpec(R)) the associated lo-

cally ringed space to R. The R-module L defines an OSpec(R)-module L, and L is a sheaf

of Lie algebroids over (Spec(R),OSpec(R)). Conversely, the algebraic version of the Serre–

Swan theorem states that any finitely generated, projective module L over R is given by

the global sections of a locally free OSpec(R)-module of finite rank. This implies that lo-

cally free Lie algebroids of constant rank over (Spec(R),OSpec(R)) correspond to finitely

generated, projective Lie–Rinehart algebras L over R, i.e. the structure maps also corre-

spond.

3. Holomorphic Lie algebroids. Given a complex manifold X, a holomorphic vector bundle

A → X is a holomorphic Lie algebroid if the sheaf of sections ΓA of A has the structure

of a sheaf of complex Lie algebras together with a holomorphic bundle map ρ : A → TX,

such that

[X1, fX2] = Lρ(X1)(f)X2 + f[X1, X2]

for all f ∈ OX(U), X1, X2 ∈ A(U) and all U ⊂ X. The pair (Γ(A),O(X)) given by the

global sections of A and the global holomorphic functions on X is a Lie–Rinehart algebra,

but it is better to view them as sheaves of Lie algebroids, since holomorphic Lie algebroids

cannot be constructed from their global sections in general.

4. Foliations. Given a smooth manifold M, an involutive subbundle F ⊂ TM, i.e. a subbun-

dle which is closed under the Lie bracket, forms a Lie algebroid with the anchor map equal

to the natural inclusion i : F → TM. By the Frobenius theorem it is integrable, which

means that M is equipped with a regular foliation such that F is given by the vector fields

that are tangent to the foliation c.f. [MoMr] for an extensive discussion. This example

extends to the holomorphic category: Given complex manifold X, an involutive subbundle

of the sheaf TX of holomorphic vector fields is a holomorphic Lie algebroid over X.




5. Action algebroids. Let g be a finite dimensional real Lie algebra equipped with an action

on M, i.e. a Lie algebra morphism a : g → Γ(TM). The trivial vector bundle g × M

admits the structure of a Lie algebroid as follows. The anchor map on constant sections

is given by the action a, and by C∞(M)-linearity this determines the anchor map on all

smooth sections Γ(g ×M). For constant sections s, t, corresponding to elements s, t ∈ g

the Lie algebroid bracket is defined as [s, t], and this determines the bracket on all smooth

sections Γ(g×M) by the Leibniz identity. This Lie algebroid is called the action algebroid

of the action of g on M and denoted by g M. Again, this example works in the

holomorphic category as well: Given a complex Lie algebra g, and a Lie algebra morphism

a : g → Γ(TX) to the space of global holomorphic vector fields, one can equip the trivial

bundle g× X in a similar fashion with the structure of a holomorphic Lie algebroid.

6. Poisson structures. Recall that a Poisson manifold is a smooth manifold P together with

a Poisson bracket , on the smooth functions C∞(P). A Poisson bracket is a bilinear

operation which is a Lie bracket and satisfies the Leibniz rule:

fg, h = fg, h+ gf, h, f, g, h ∈ C∞(P).

This bracket is equivalent to a bivector field Π ∈ Γ(Λ2TP) characterized by Π(df, dg) =

f, g, which satisfies [Π,Π]SN = 0. The Poisson structure defines a Lie algebroid structure

on T∗P as follows: The anchor map ρ is contraction by the Poisson tensor, i.e.

ρ(α)(β) = Π(α,β), α, β ∈ Ω1(P)

and the algebroid bracket is given by

[α,β]T∗P := Lρ(α)(β) − Lρ(β(α) − d(Π(α,β)).

Locally, i.e. when we can write α = df and β = dg, the anchor and brackets are given by

ρ(df) = Xf [df, dg]T∗P = f, g.

where Xf is the Hamiltionian vector field associated to f with respect to the Poisson

structure, and in fact these equations determine the bracket globally. Given a complex

manifold X, a holomorphic Poisson structure is defined as a Poisson structure on the sheaf

of holomorphic functions, i.e. for each open U ⊂ X the algebra OX(U) of holomorphic

functions is a Poisson algebra. It is characterized by a global section of the sheaf of

holomorphic biderivations of X, which we denote by Γ(Λ2TX). The formulas that define a

Lie algebroid structure on T∨X , the sheaf of holomorphic one forms on X, carry over from

the smooth case without adaptations.

7. Atiyah algebroids. Given a smooth vector bundle E → M, the subspace Diff1(Γ(E)) ⊂EndR(Γ(E)) of order 1 differential operators is defined by

h ∈ Diff1(Γ(E)) ⇔ [h, f] ∈ End(E) ∀f ∈ C∞(M).

The map f → [h, f] is a derivation and thus it defines an element σ(h) ∈ TM ⊗ End(E).

It defines the symbol map

σ : Diff1(Γ(E)) −→ TM⊗ End(E), h → σ(h),

which is a C∞(M)-linear map of vector bundles with kernel End(E). A TM-connection

(or affine connection) on E, which we will define in the next subsection, defines a splitting

of the exact sequence

0 −→ End(E) −→ Diff1(Γ(E)) −→ TM⊗ End(E) −→ 0




by the formula X ⊗ e → ∇Xe where X ⊗ e ∈ TM ⊗ End(E), and this gives the middle

C∞(M)-module the structure of a vector bundle. The Atiyah algebroid At(E) is defined

as the subbundle of Diff1(Γ(E)) given by differential operators with scalar symbol, i.e.

it fits into the exact sequence

0 −→ End(E) −→ At(E) −→ TM −→ 0. (1.1.1)

The anchor map is provided by the sequence above, and the Lie bracket is given by the

commutator of differential operators. It is easy to check that the commutator of two

operators with scalar symbol again has a scalar symbol, so this is well-defined. Again,

the definition in the holomorphic category is the same; then the Atiyah algebroid is given

as the sheaf of order 1 differential operators with scalar symbol.

8. Lie algebroids associated with hypersurfaces. Let N be a hypersurface in smooth manifold

M. Let Γ(TM(− logN)) be the space of global vector fields that are tangent to N. It is

a C∞(M)-module which acts by derivations on C∞(M), and one can prove, c.f. [Me] for

a proof in the case of manifolds with boundary, that it is projective, i.e. it is given by

the sections of a vector bundle. We denote this vector bundle by TM(− logN). The Lie

bracket is given by the commutator of vector fields and the anchor map is given by the

inclusion. Let (x, y2, . . . yn) be local coordinates on M such that N is defined by x = 0.

Then it is clear that

x∂

∂x,

∂

∂y2, . . . ,

∂

∂yn

forms a local basis for TM(− logN). Hence, when x > 0, the image of x∂/∂x under the

anchor map evaluates to 1 in the differential form d(log(x)).

In the holomorphic setting the definition is as follows. LetD ⊂ X be a smooth holomorphic

hypersurface. The OX-module TX(− logD) is defined as the sheaf of sections of TX which

preserve functions that vanish on D, i.e. when f is a function such that f|D = 0, then

V ∈ TX(− logD) iff V(f)|D = 0. When D is not smooth, the definition still applies and it

gives an involutive subbundle of TX, however it will not be locally free in general, see [P]

for more details. The condition that D has normal crossing singularities, i.e. one can find

coordinates x1, . . . , xk around singularities of D such that D is defined by x1x2 · · · xk = 0,

implies that TX(− logD) is locally free.

9. Lie algebroids associated to Lie groupoids. In this example we discuss Lie groupoids,

which can be viewed as generalized, global symmetries of spaces. First we give the def-

inition of a smooth Lie groupoid. We refer to [M] for more details and many examples

of Lie groupoids. Replacing all the adjective ‘smooth’ by ‘holomorphic’ in the following

definition defines the notion of a holomorphic Lie Groupoid. Moreover, one can associate

a holomorphic Lie algebroid to a holomorphic Lie groupoid, and the examples that we

discuss are also defined in the holomorphic category.

Definition 1.1.7. A Lie groupoid is given by a pair of manifolds (G1, G0) and a collection

of smooth maps

s, t : G1 → G0, u : G0 → G1, inv : G1 → G1, m : G1s×tG1 → G1.

called the source map s, the target map t, the unit map u, the inverse inv and the mul-

tiplication m. The source and target map are submersions (hence G1s×tG1 is a smooth

manifold). Moreover, the pair (G1, G0) is given by the set of arrows and the set of objects

of a category in which all arrows are invertible, and the structure maps of this category

coincide with s, t, u, inv and m.




We remark that G1 is not required to be Hausdorff, as opposed to G0 and s and t fibers.

Note that the fibers of s and t are smooth manifolds. It is customary to write G1 ⇒ G0

to denote a Lie groupoid and to call points of G1 arrows and points of G0 objects.

To each Lie groupoid one can associate a Lie algebroid over G0 which encodes much of the

information of the Lie groupoid. The construction is as follows: We denote the subbundle

ker(Ts) ⊂ TG which fiber consists of vectors in TG which are tangent to the s-fibers by

TsG, and claim that

id∗(TsG),

the pull-back bundle of TsG1 under the map id, is a Lie algebroid over G0. An element

g ∈ G1 defines a smooth diffeomorphism of fibers:

Rg : s−1(y) −→ s−1(x)

where x = s(g) and y = t(g). Because it is a diffeomorphism, it induces an isomorphism

TRg : Tsh(s

−1(y)) −→∼= Tshg(s

−1(y))

where s(h) = y. It follows that a section X ∈ Γ(id∗(TsG1)) can be uniquely extended to

a right invariant vector field Xinv in TsG1 by the formula

Xinvg := TRg(Xid(y))

The Lie bracket of two invariant vector fields on G1 is again an invariant vector field,

hence one obtains a well-defined bracket on sections of id∗(TsG1). The anchor map is

given by ρ = Tt. More precisely, given a section X ∈ Γ(id∗(TsG1)), the anchor is given by

Tt(Xinv), which is well-defined since Xinv is invariant. The fact that it satisfies the Leibniz

identity follows from the Leibniz identity for vector fields on G1.

10. Some concrete examples of Lie groupoids. LetM be a smooth manifold. The pair groupoid

is defined as G1 ⇒ G0 = M × M ⇒ M. The source and targent maps s and t are the

left and right projections, the identity id is given by diagonal inclusion M → M×M, the

multiplication m is defined by m((x, y), (y, z)) = (x, z) and the inverse inv by inv(x, y) =

(y, x). Its associated Lie algebroid is TM with the standard structure maps.

Let M be a smooth manifold and F an involutive subbundle of TM. The following pro-

cedure associates a Lie groupoid, called the monodromy groupoid, Mon(M,F) ⇒ M to

F. Points of Mon(M,F) (arrows) are given by homotopy classes of paths with fixed end

points which map into leaves of the foliation. The multiplication m is given by concatena-

tion of paths, the source and target send a path to its starting respectively end point, the

inverse reverses paths and the inclusion sends a point to a constant path in that point.

The Lie algebroid which is associated to Mon(M,F) ⇒ M is given by F. See [MoMr] for

a discussion.

Let a : G × M → M be a smooth action of a Lie group G on a manifold M. Set

G × M = G1 and G0 = M and define s(g, x) = x , t(g, x) = a(g, x) , id(x) = (e, x) ,

inv(g, x) = (g−1,m) and m((gx, ghx)(x, gx) = (x, ghx)) where e ∈ G is the identity.

Then G1 ⇒ G0 is a Lie groupoid called the action groupoid of the action of G on M. It

is denoted by GM ⇒ M, and its associated Lie algebroid is the action algebroid of the

action of g on M obtained by differentiating the action of G on M, where g is the Lie

algebra of G.

11. A remark about integrability A Lie groupoid G1 ⇒ G0 with associated algebroid A is

called an integration of A. We have seen that the tangent bundle and Lie algebroids with

an injective anchor map always admit an integration, and Lie’s third theorem provides




the result in the case of Lie algebras. In general it is not possible to find integrations

of Lie algebroids. The obstructions to integrability were determined [CF1], which also

provides an extensive overview of the subject.

1.1.2 Alternative definitions of Lie algebroids

It is often possible to generalize the differential geometric constructions that are performed

with tangent bundles to the realm of Lie algebroids, and moreover several notions of Lie algebra

theory can be extended to Lie algebroids. We will start with constructions that can be performed

in the greatest generality, namely that of sheaves of Lie algebroids and discuss particular cases

later.

The complex of L-de Rham forms

Let (X,OX) be a ringed space and L a sheaf of Lie algebroids over (X,OX)

Definition 1.1.8. The de Rham complex of L-forms is a sheaf of differential graded algebras

(dg algebras) given by

Ω•L := Λ•

OXL∨

with the product given by the wedge product ∧ and the differential dL defined in low degrees by

dLf(l1) :=l1(f)

dLα(l1, l2) :=l1(α(l2)) − l2(α(l1)) − α([l1, l2]),

where f ∈ OX, li ∈ L and α ∈ L∨, and extended by the derivation property.

Conversely, one can show that a differential of degree 1 which acts by derivations on Ω•L

defines a structure of a Lie algebroid on L, see for example [Xu99]. Given three OX-modules

E,F and G together with an OX-linear operation

E⊗OXF −→ G

it follows easily that one can define an operation

Ω•L(E)⊗OX

Ω•L(F) −→ Ω•

L(G)

where Ω•L(E) := Ω•

L ⊗OXE. One of the simplest examples is the OX-module structure on

E itself, which equips Ω•L(E) with the structure of an Ω•

L-module. Other examples are the

composition of endomorphisms EndOX(E), the action of endomorphisms EndOX

(E) on E and

the multiplication m for a sheaf of algebras A.

Definition 1.1.9. An L-connection on a OX module E is a K-linear map

∇ : E −→ L∨ ⊗OXE

satisfying the Leibniz rule

∇(fs) = dLf⊗ s+ f∇(s), f ∈ OX(U), s ∈ E(U)

for all sections and opens U ⊂ X.

As remarked earlier, the OX-module structure on E defines an Ω•L-module structure on

ΩL(E), and the formula

d∇(α⊗ s) = dL(α)⊗ s+ (−1)|α|α∧∇(s)




defines a map d∇ of degree 1 on Ω•L(X,E), which is more explicitly given by the Koszul formula

d∇ω(l1, . . . , lk+1) :=

k+1∑i=1

(−1)i+1∇li(ω(l1, . . . , li, . . . , lk+1))

+

k+1∑i,j=1

(−1)i+jω([li, lj], l1, . . . , li, . . . , lj, . . . , lk+1).

(1.1.2)

One can show that d2∇ = R∇, where R∇ ∈ Ω2

L(End(E)) is the curvature of the connection,

defined by

R∇(l1, l2) := [∇l1 ,∇l2 ] −∇[l1,l2], li ∈ L.

If R∇ = 0, ∇ is flat and defines a representation of L on E. The complex (Ω•L(E), d∇) for a

flat connection ∇ is called the complex of L de Rham forms with values in E.

Example 5. The action of L on OX defines a flat representation and the resulting complex is

just the de Rham complex (Ω•L, dL).

The L-polyvectorfields.

We start by recalling the definition of a Gerstenhaber algebra:

Definition 1.1.10. A Gerstenhaber algebra A is a graded vector space equipped with a Lie

bracket [ , ] of degree −1 and a (super)commutative, associative product · of degree 0 such that

[α,β · γ] = [α,β] · γ+ (−1)|α|(|β|−1)β · [α, γ]

for all α,β, γ ∈ A.

Dual to the L-de Rham forms one has the sheaf of L-polyvector fields defined by

TLpoly := Λ•

OXL.

The L-polyvectorfields are endowed with the graded commutative wedge product ∧, and the

Lie bracket on L can be extended to a bracket on L–polyvectorfields by the following definition:

[l1∧ · · ·∧ lm, k1∧ · · ·∧kn] :=∑i,j

(−1)i+j[li, kj]l1∧ · · · li · · ·∧ lm∧k1∧ · · · kj · · ·∧kn (1.1.3)

for all li, kj ∈ L, and

[f, l1 ∧ · · ·∧ lm] = −ιdLfl1 ∧ · · ·∧ lm

where f ∈ OX, and ιdLf is defined by

ιdLf(l1 ∧ · · ·∧ lm)(α1, . . . , αm−1) := l1 ∧ · · ·∧ lm(dLf, α1, . . . , αm−1)

for αi ∈ L∨. The bracket is called the Schouten–Nijenhuis bracket and denoted by [ , ]SN or

sometimes simply by [ , ]. In low degrees the bracket is given by

[l1, f]SN :=l1(f)

[l1, l2]SN :=[l1, l2]

for f ∈ OX and li ∈ L and the full bracket can also be defined as the extension of this bracket

using the Leibniz rule:

[γ1, γ2 ∧ γ3]SN := [γ1, γ2]SN ∧ γ3 + (−1)|γ1|(|γ2|+1)γ2 ∧ [γ1, γ3]SN

where γi ∈ TLpoly are homogeneous elements and | | denotes the degree of polyvectorfields.




Proposition 1.1.11. The triple (TLpoly,∧, [ , ]SN) is a sheaf of Gerstenhaber algebras.

Given an OX-module L, a structure of a sheaf of Gerstenhaber algbras on the sheaf Λ•OX

L

determines a Lie algebroid structure on L on OX. It follows that

Proposition 1.1.12. Given a locally free OX-module L on a ringed space (X,OX), the following

three structures are equivalent:

1. The structure of a sheaf of Lie algebroids on L.

2. A bracket [ , ]SN for which (TLpoly,∧, [ , ]SN) is a sheaf of Gerstenhaber algebras.

3. A degree 1 differential dL for which (ΩL,∧, dL) is a sheaf of dg algebras.

A proof for the smooth case can be found in [Xu99]; the argument extends easily to the

more general case.

Definition 1.1.13. Given a Gerstenhaber algebra (A,∧, [ , ]), a precalculus over A is given

by a graded vector space M and operations ι and L such that:

1. The operation

ι : A⊗M −→ M

gives M the structure of a graded module over (A,∧).

2. The operation

L : A⊗M −→ M

gives M the structure of a graded Lie module over (A[1], [ , ]).

3. The following compatibilities hold for all a, b ∈ A:

[La, ιb] = ι[a,b] ιbLa + (−1)|b|Laιb = La∧b

where we used the graded commutators on the LHS.

A precalculus is a calculus when there exists a derivation d of degree −1 on M such that

[d, ιa] = (−1)|a|−1La.

The natural contraction operation between L-differential forms and L-polyvectorfields is given

by

ιγ1ω(γ2) := ω(γ1 ∧ γ2), ω ∈ ΩL, γ1, γ2 ∈ TL

poly

Proposition 1.1.14. The de Rham differential dL on L-forms turns (ΩL, ι, dL, L) with the

negative grading into a sheaf of calculi over the sheaf of Gerstenhaber algebras (TLpoly,∧, [ , ])

The proof of this proposition in the smooth case can be found in [Xu99] or [ELW], although

there the notion of a calculus is not mentioned. Again it extends to the general case without

difficulties.

Remark 1.1.15. The latter structure is, for this thesis, merely a convenient way to summarize

all the structures on the L-forms and L-polyvectorfields that we will use.




1.1.3 Lie algebroid cohomology

Given a Lie algebroid L over (X,OX), the cohomology of L is given by the hypercohomology of

the complex of sheaves

OXdL−→ Ω1

L

dL−→ Ω2L

dL−→ . . .

which we denote by H•(L) = H•(Ω•L, dL). As the case of Lie algebras already shows, this

complex is not locally acyclic. In the smooth case, this implies that the complex is in general

not a resolution of fine sheaves of the sheaf of constant functions.

Example 6.

1) When L is smooth, the sheaves Ω•L are fine. Hence, the cohomology can be computed as

the cohomology of the complex of global sections of the sheaves.

2) Let (L, R) be a Lie–Rinehart algebra such that L is projective over R. One can define

the complex of differential forms as HomR(Λ•L, R) and the differential by the standard Koszul

formula. Note that Λ•L∨ is not the right notion if L is not finitely generated. In [Rin], Rinehart

showed that the cohomology H•(L, R) of this complex is equal to ExtUL(R, R), where UL is the

universal enveloping algebra (see §1.2.1), i.e. the cohomology theory forms a part of homological

algebra.

3) For a holomorphic Lie algebroid L, the Dolbeault resolution of [LSX12] can be applied. We

discuss this below. An alternative is provided by the standard Cech resolution of sheaves which

also applies in the more general setting of sheaves of Lie algebroids over ringed spaces.

Given an L-module E, i.e. an OX-module with a flat L-connection, the hypercohomology

H•(Ω•L(E), d∇) of the complex

OXd∇−→ Ω1

L(E)d∇−→ Ω2

L(E)d∇−→ . . .

is defined to be H•(L,E), the cohomology of L with values in E.

Example 7.

1) Let L = TM for M a smooth manifold. Representations of TM are given by flat vector

bundles E over M. The cohomology with values in E is given by the usual cohomology of M

with local coefficients given by the flat sections of E. In particular, when E is the trivial line

bundle, one recovers the de Rham cohomology of M.

2) When A = g, a Lie algebra over a point, representations of A are the same as Lie algebra

representations of g, and the algebroid cohomology agrees with the Lie algebra cohomology

obtained from the Chevalley–Eilenberg complex. A natural example, besides the trival repre-

sentation, is the adjoint representation of g, which leads to a cohomology theory controlling

deformations of g. In chapter 3 we will discuss a broader notion of representations that encom-

passes the adjoint representation for any smooth Lie algebroid.

Since it will be used in chapter 4, we discuss the example of a holomorphic Lie algebroid in

more detail:

Example 8. The Dolbeault resolution for a holomorphic Lie algebroid A → X over a complex

manifold X was constructed in [LSX08]. Similar to the holomorphic tangent bundle, we consider

the Lie algebroid A as a smooth Lie algebroid AR together with an almost complex structure,

i.e. a bundle map

J : AR −→ AR

which squares to −1 and is compatible with the anchor map and the bracket. The complexified

Lie algebroid AR⊗C splits into AR⊗C ∼= A1,0⊕A0,1 according to the ±i eigenvalues of J, and

(A1,0, A0,1) forms a matched pair, i.e. a pair of Lie algebroids such that A1,0 is an A0,1-module



1.2. NONCOMMUTATIVE GEOMETRY OF LIE ALGEBROIDS 21

and A0,1 is an A1,0-module. The module structures and the anchor map need to satisfy a

number of compatibility relations, see [LSX08].

It is a well-known fact that a holomorphic structure on a vector bundle is equivalent to a

flat T0,1X-connection on the bundle (the holomorphic sections are given by flat sections), hence

A1,0 is a representation of T0,1X. The formula

∇s(X) = pr1,0([ρ(s), X]), s ∈ A1,0, X ∈ T0,1X

gives T0,1X the structure of anA1,0-representation, and these two representation give (T0,1X, A1,0)

the structure of a matched pair.

A crucial property of matched pairs is that their direct sum admits the structure of a Lie

algebroid, which is, in the case at hand, denoted by T0,1X A1,0. The cohomology of the

latter Lie algebroid is equal to the cohomology of the total complex associated to the following

double complex:. . . . . .

D1,0A D1,1

A . . .

D0,0A D0,1

A . . .

d1,0A

∂

d1,0A

∂

∂

d1,0A d1,0

A

∂

where Dp,qA := ∧p(A1,0)∨ ⊗ ∧q(T0,1X)∨, and the differentials are induced by the module

structures. In this particular case the flat T0,1X -connection on A1,0 gives rise to the Dolbeault

differential ∂ for A, hence the ∂-Poincare lemma shows that the double complex above is a

resolution of

OXdA−→ Ω1

A

dA−→ Ω2A

dA−→ . . .

and one has:

Theorem 1.1.16 ([LSX08]). Let A be a holomorphic Lie algebroid. Then we have

1. The holomorphic de Rham isomorphism: H•(A) = H•(T0,1X A1,0), where the LHS is

the hypercohomology of the complex of holomorphic de Rham forms, and the RHS is the

cohomology of the smooth Lie algebroid T0,1X A1,0.

2. The Dolbeault isomorphism: Hq(ΩpA) = Hq(Dp,•

A , ∂), where the LHS is the sheaf coho-

mology of ΩpA(X).

1.2 Noncommutative geometry of Lie algebroids

The universal enveloping algebra of a Lie algebra

First we describe the construction of a universal enveloping algebra associated to a Lie algebra

g over a field K ⊃ Q. The universal enveloping algebra of a Lie algebra g is defined as quotient

of the tensor algebra:

U(g) := T(g)/I

where I is the ideal generated by the set i(x) ⊗ i(y) − i(y) ⊗ i(x) − i([x, y]), x, y ∈ g and

i : g → T(g) is the natural inclusion. The universal property of the universal enveloping

algebra is a bijection HomLie(g,Lie(A)) HomAss(U(g), A), where Lie(A) is A equipped with

the commutator bracket. When the Lie bracket on g is trivial, the universal enveloping algebra

of g is the symmetric algebra of g, denoted S(g). The symmetric algebra can also be realized

as the subalgebra of the tensor algebra consisting of symmetric tensors.




The natural filtration on the tensor algebra descends to the universal enveloping algebra,

denoted by U(g)0 ⊂ U(g)1 ⊂ . . .. Let Gr(U(g)) be its associated graded algebra: Gr(U(g)) :=⊕∞i=0 U(g)i/U(g)i−1

Theorem 1.2.1 (Poincare–Birkhoff–Witt). The composition of the maps

S(g) −→ T(g) −→ U(g) −→ Gr(U(g))

given by the symmetrization map

x1 · · · xk → 1

k!

∑σ∈Sk

xσ(1) ⊗ · · · ⊗ xσ(k) (1.2.1)

and the natural projections, is an isomorphism of graded algebras.

A proof can be found in [Se]. Because its associated graded algebra is commutative, the uni-

versal enveloping algebra is called almost commutative. The associated graded algebra inherits

a Poisson structure from the commutator bracket on U(g) as follows: Let x ∈ Gri(U(g)) and

y ∈ Grj(U(g)), and choose representatives x ∈ U(g)i and y ∈ U(g)j. We have [x, y] ∈ Ui+j−1(g)

because U(g) is almost commutative. Then the Poisson bracket between x and y is defined as

the class of [x, y] in Gri+j−1(U(g)), and it is straightforward to show that this definition does

not depend on the choice of lifts.

1.2.1 The universal enveloping algebra of a Lie algebroid

Let L be a sheaf of Lie algebroids over a ringed space (X,OX). The sheaf OX ⊕ L is a sheaf of

Lie algebras with the bracket defined by

[(f1, l1), (f2, l2)] := (l1(f2) − l2(f1), [l1, l2])

for fi ∈ OX and li ∈ L. The sheaf of universal enveloping algebras of this sheaf of Lie algebras

has a subsheaf which is generated by i(O ⊕ L), and the quotient of this sheaf by the relation

f ⊗ (g ⊕ l) = fg ⊗ fl for f ∈ OX and l ∈ L is defined to be the sheaf of universal enveloping

algebras, sometimes abbreviated by universal enveloping algebra, and denoted by U(L).

The universal enveloping algebra is unital and is equipped with a K-algebra morphism

i : OX → U(L), and a morphism i : L → Lie(U(L)) of K-Lie algebras, subject to the conditions

i(f)i(X) = i(fX), i(X)i(f) − i(f)i(X) = i(X(f)), f ∈ OX, X ∈ L.

The universal enveloping algebra satisfies the following universal property: for any triple

(A, φL, φO) consisting of a sheaf of K-algebras A, a homomorphisms φOX: OX → A, and

a Lie algebra morphism φL : L → Lie(A), such that for all f ∈ OX, l ∈ L

φOX(f)φL(X) = φL(fX), φL(X)φOX

(f) − φOX(f)φL(X) = φOX

(X(f)),

there is a unique morphism Φ : U(L) → A of sheaves of K-algebras such that Φ iO = φ :

OX → U(L) and Φ iL = φ : L → U(L).

Recall that a locally free OX-module E is an (OX,L)-module iff there exists a flat L-

connection on E. The map L → EndK(E) given by X → ∇X exactly satisfies the universal

property given above, hence the natural functor

Mod(U(L)) → Mod(OX,L) (1.2.2)

is an equivalence of categories.




Example 9.

1) Given a smooth manifold M, the universal enveloping algebra of TM is given by the sheaf of

differential operators Diff(M) on M. Likewise, the universal enveloping algebra of the holomor-

phic tangent bundle TX of a complex manifold X is given by the sheaf of holomorphic differential

operators on X.

2) For G ⇒ M a Lie groupoid with associated Lie algebroid A → M, the universal enveloping

algebra of A is a model for the algebra of differential operators which are tangent to the s-fibers

invariant under the right action on the s-fibers. For differential operators of order 1 this is the

correspondence between invariant fiberwise vector fields and the Lie algebroid A.

3) Given a regular foliation of a smooth manifold M with associated Lie algebroid A, the uni-

versal enveloping algebroid of A is given by the leafwise differential operators on M, i.e. the

subalgebra of differential operators on M which are tangent to the foliation.

1.2.2 Hochschild and Cyclic theory

Basic definitions

We briefly recall the basic notions of Hochschild and cyclic homology as well as their dual

cohomology theories of an associative, unital algebra C, to which we will refer by the cyclic

theory of C. Then we give a few examples. An extensive introduction into the subject is given

in [Lo]. We fix a unital algebra C over a field K which is either R or C. The fact that C is

unital allows for considerable simplifications of the definitions, and in this thesis all the algebras

that we consider are unital. Let M be an C-bimodule and write C := C/1 ·K. If C and M are

graded, one has to apply the Koszul sign rule in all the definitions that follow. Set

C•(C;M) := M⊗ C⊗•

C•(C;M) := HomK(C⊗•,M).

Let ai ∈ C and m ∈ M. As one can easily check, the formula

bH(m⊗a1 ⊗ . . .⊗ ak) := m · a1 ⊗ a2 ⊗ . . .⊗ ak

+

k−1∑i=1

(−1)im⊗ a1 ⊗ . . .⊗ aiai+1 ⊗ . . .⊗ ak + (−1)kak ·m⊗ a1 ⊗ . . .⊗ ak−1

defines a differential bH of degree −1 on C•(C;M), and the formula

bHφ(a1, . . . , ak) :=a1φ(a2, . . . , ak)

+

k−1∑i=1

(−1)iφ(a1, . . . , aiai+1, . . . , ak) + (−1)kφ(a1, . . . , ak−1)ak

endows C•(C;M) with a differential bH of degree 1. The resulting homology of the first complex

is called the Hochschild homology of C with values in M, written H•(C;M), and the second

complex defines the Hochschild cohomology H•(C;M) of C with values in M. When M = C,

the Hochschild homology complex is denoted by C•(C) := C•(C,C) and the Hochschild homol-

ogy by HH•(C) =: H•(C;C). The Hochschild cohomology H•(C;C) is called the Hochschild

cohomology of C and denoted by HH•(C). The formula

B(a0 ⊗ . . .⊗ ak) :=

k∑i=0

(−1)ik1⊗ ai ⊗ a . . .⊗ ak ⊗ a0 ⊗ . . .⊗ ai−1

defines a differential of degree 1 on the Hochschild chain complex C•(C;C). Besides the relation

B2 = 0 it satisfies [b, B] = 0. The dual of B, which we also denote by B, defines a differential

of degree −1 on C•(C,C∗), the Hochschild cochain complex with values in the dual of C. To




define the chain complexes for the various versions of cyclic homology, we introduce, following

[GJ], a formal variable u of degree 2 and a formal variable u−1 of degree −2 and set

CCW• (C) := C•(C;C)[[u

−1|]]⊗K[u−1] W

CC•W(C) := C•(C;C∗)[[u]]⊗K[u] W.

and equip them with the differentials bH + u−1B and bH + uB. Here, W denotes a K[u−1]-

module in the first line and a K[u]-module in the second. Various choices for W lead to the

following theories, of which the last three are homology theories.

1. W = K gives the Hochschild homology of C and the Hochschild cohomology with values

in C∗.

2. W = K((u−1))/u−1K[[u−1]] = K[u] leads to (ordinary) cyclic homology, denoted by

HC•(C).

3. W = K((u−1)) leads to periodic cyclic homology, denoted HP•(C). It has period 2 with

periodicity operator induced by multiplication by u.

4. W = K[u−1] leads to negative cyclic homology, denoted HC−• (C).

Short exact sequences of K[u]-modules lead to long exact sequences in homology. For example,

the usual SBI-sequence is induced from the short exact sequence

0 −→ K((u−1))/K[[u−1]] −→ K((u−1))/u−1K[[u−1]] −→ K −→ 0.

On the other hand, the short exact sequence

0 −→ u−1K[u−1] −→ K((u−1)) −→ K((u−1))/u−1K[[u−1]] −→ 0

leads to the long exact sequence

. . . −→ HC−n+2(C) −→ HPn(C) −→ HCn(C) −→ HC−

n+1 −→ . . .

relating negative and periodic cyclic homology.

Additional structures on the Hochschild (co)complex

We define a few operations on the Hochschild and cyclic (co)chain complexes that we will use

in chapter 4. They are part of the set of operations which defines the structure of a calculus on

the associated (co)homology complexes, c.f. [DTTs]. Let a ∈ C.

1. The insertion operator ιa : C•(C) → C•+1(C) is defined by

ιaa0 ⊗ · · · ⊗ ak :=

k∑i=0

(−1)i+1a0 ⊗ · · · ⊗ ai ⊗ a⊗ ai+1 ⊗ · · · ⊗ ak).

2. Let Lie(C) be the Lie algebra given by C equipped with the commutator bracket. The

formula

La = [b, ia] (1.2.3)

equips C•(C) with a structure of a Lie algebra module over Lie(C). Explicitly, it is given

by the following formula:

La(a0 ⊗ · · · ⊗ ak) =

k∑i=0

(a0 ⊗ · · · ⊗ [a, ai]⊗ . . . , ak).

The dual formulas induce actions on C•(C,C∗).




3. The shuffle product on the Hochschild chain complex is defined by the formula:

(a0 ⊗ · · · ⊗ ap)× (b0 ⊗ · · · ⊗ bq) = a0b0 ⊗ Shp,q(a1 ⊗ · · · ⊗ ap ⊗ b1 ⊗ · · · ⊗ bq),

where a0 ⊗ · · · ⊗ ap ∈ Cp(C) and b0 ⊗ · · · ⊗ bq ∈ Cq(C) and Sp,q ⊂ Sp+q consists of the

p, q shuffle permutations in the group of permutations Sp+q.

Remark 1.2.2. Recall that Hochschild (co)homology can be computed using methods of ho-

mological algebra. Given an algebra C as above, and an C-bimodule, one has:

H•(C,M) ∼= TorCe

• (M,C)

H•(C,M) ∼= Ext•Ce(C,M).

Here, Ce := C⊗Cop is the enveloping algebra of C, which has the property that left Ce-modules

are equivalent to C-bimodules. A proof can be found in [We]. As we will see in the examples,

sometimes it is convenient to find the appropriate projective or injective resolution to compute

the Hochschild (co)homology.

Example 10. (Algebraic HKR theorem) In this example we discuss the Hochschild homology

of a commutative, unital algebra C over a field K such that Q ⊂ K. We will see that it is given

by the algebraic de Rham forms, which indicates that the Hochschild homology can serve as a

model for de Rham theory for noncommutative algebras.

The C-module of algebraic de Rham forms of degree 1 of a commutative algebra, denoted

by Ω1C|K

, is defined to be the C-module presented by elements da where a ∈ C, subject to the

relations

dλ = 0 d(a1 + a2) = da1 + da2 d(a1a2) = a1da2 + a2da1

for all λ ∈ K and ai ∈ C. They are also referred to by Kahler differentials. The algebraic de

Rham forms in degree n are defined as

ΩnC|K

:= ΛnCΩ

1C|K

and the natural derivation

C → Ω1C|K

, a → da,

can be uniquely extended to a derivation d of degree 1 on Ω•C|K

given by

a0da1 · · ·dn → da0da1 · · ·dan

which satisfies d2 = 0. The resulting complex is called the algebraic de Rham complex of C

and the cohomology of this complex is the algebraic de Rham cohomology of C, denoted by

H•dR(C). There are two natural maps relating the Hochschild chain complex and the complex

of algebraic de Rham forms:

εn : ΩnC|K

−→ Cn(C) εn(a0da1 · · ·dan) :=∑σ∈Sn

(−1)|σ|a0 ⊗ aσ(1) ⊗ · · · ⊗ aσ(n)

µn : Cn(C) −→ ΩnC|K

µ(a0a1 ⊗ · · · ⊗ an) :=1

n!a0da1 · · ·dan.

One can show that the map µn defines a morphism of complexes

µ : (CCW• (C), b+ u−1B) −→ (ΩW

C|K, 0+ u−1d)

where ΩWC|K

= ΩC|K ⊗K[u−1] W. If the algebra C is smooth, see [Lo] for the precise definition,

one has the following theorem.




Theorem 1.2.3 ([HKR, LoQ]). If C is a smooth algebra over K, the map ε induces an iso-

morphism

Ω•C|K

−→ H•(C).

Moreover, if Q ⊂ K, one has canonical isomorphisms

HC•(C) ∼= Ω•C|K

/dΩ•C|K

⊕H•−2dR (C)⊕H•−4

dR (C)⊕ . . .

HP•(C) ∼=⊕i∈Z

H•+2idR (C)

where HC• stands for cyclic homology and HP• stands for periodic cyclic homology.

Example 11. (Connes’ theorem.) When the algebra C admits a topology which is compatible

with the algebra structures, the Hochschild complex is defined using a different version of the

tensor product, which takes the topology into account. This is done because the cyclic theory

is rather hard to compute with the algebraic definitions, e.g. in the case of smooth functions

on a manifold M. In this case, the Hochschild complex can be defined as

C•(C∞(M)) =C∞(M×•)

but also as the complex of germs or even jets of functions on the diagonals in M×•. Connes

proved the following theorem.

Theorem 1.2.4 ([Co]). Let M be a smooth compact manifold and let Ωk(M) be space of smooth

k-forms on M. Then one has isomorphisms

HH•(C∞(M)) ∼= Ω•(M)

HP•(C∞(M)) ∼= Heven

dR (M)⊕HodddR (M).

Dualizing both complexes yields the space of de Rham currents, i.e. the continuous linear

functionals on the de Rham forms, and the Hochschild cohomology with values in the continuous

dual C∞(M)∗ respectively. The dualized morphism of complexes µ∗ is a quasi-isomorphism on

these complexes. The extension of the HKR theorem to multivector fields will be discussed later,

and we also define other, smaller, Hochschild complexes, namely the polydifferential operators

and jets. In fact, the HKRC theorem is proved by considering a stable subcomplex of the given

Hochschild complex consisting of jets along the diagonal M ⊂ M×k.

Example 12 (Almost commutative algebras). This computation was done in [Ka] by Kassel.

Let C be an associative, unital algebra with a filtration . . . ⊂ FnC ⊂ Fn+1C ⊂ . . . such that

∪FnC = C and ∩ FCn = 0.

The algebra C is called almost commutative if the associated graded algebra

Gr(C) :=⊕n+1

FnC/FnC

is isomorphic to the symmetric algebra Sym(V) of V, where V = F1C/F0C. As in the case of

the universal enveloping algebra of a Lie algebra, Gr(C) is a Poisson algebra. The bracket is

defined by the same formula:

f, g := σm+n−1([a, b]),

where σn is the projection σn : FnC → FnC/Fn−1C and a and b are lifts of f and g, i.e.

σ(a) = f and σ(b) = g.

The algebraic de Rham forms of the algebra Sym(V) are given by

Ω•Sym(V)

∼= Sym(V))⊗Λ•V,




and in [Br, L] it was observed that the Poisson structure on Sym(V) can be used to define the

Poisson homology differential δ of degree −1 on the de Rham complex:

δ(f0df1 . . . dfn) =

n∑i=1

(−1)i+1f0, fidf1 . . . fi . . . dfn

+∑

1i<jn

(−1)i+jf0dfi, fjdf1 . . . dfj . . . dfj . . . dfn.

This differential anti commutes with the de Rham differential d, hence one can form a mixed

complex(Ω•,W

Sym(V), δ, u−1d

). This mixed complex is related to the cyclic theory of C by the

following theorem:

Theorem 1.2.5. For C an almost commutative algebra one has the following isomorphisms:

H•(C) ∼= H•(Ω•Sym(V), δ)

HC•(C) ∼= HC•(Ω•Sym(V), δ, u

−1d)

HP•(C) ∼= HP•(Ω•Sym(V), δ, u

−1d).

Remark 1.2.6.

1) The universal enveloping algebra U(g) of a Lie algebra g and the Weyl algebra, which is

defined in appendix §B.1, are particular examples of almost commutative algebras. This gives

an easy way to determine the cyclic theory of the Weyl algebra, because the associated Poisson

structure is symplectic. The computation of the RHS in the case of the universal enveloping

algebra is more intricate.

2) The essential part of the proof that Kassel gave is the observation that the following formula

defines a map of complexes

φ : (Ω•S(g), δ) → (C•(U(g)), bH) φ(Pdx1 . . . dxn) =

∑σ∈Sn

(−1)|σ|η(P)xσ(1) ⊗ . . .⊗ xσ(n),

where j is the symmetrization map of (1.2.1) induces a quasi isomorphism of complexes. In

chapter 3, we will construct a similar map in the case a smooth Lie algebroid using the formality

theorem that Kontsevich proved in [K].






Chapter 2

The Poincare–Birkhoff–Witt

theorem

The core of this chapter is the dual Poincare–Birkhoff–Witt theorem for sheaves of Lie alge-

broids. Recall from the previous chapter that the PBW theorem for Lie algebras states that

gr(U(g)) ∼= Sym(g), and that, if Q ⊂ K, one can define the symmetrization map Sym(g) → U(g)

inducing an isomorphism on the associated graded spaces. In [Rin], Rinehart proved that

gr(U(L, R)) ∼= SymR(L) for a Lie–Rinehart pair (L, R) under the condition that L is projective

over R. However, he did not construct a map SymR(L) −→ U(L, R) inducing this isomorphism.

In [NWX] the authors used the concept of a local integrating groupoid to define a local expo-

nential map, which induced a lift

Sym(A) −→ U(A)

of the PBW theorem in the case of a smooth Lie algebroid A → M. In this chapter we generalize

this result to locally free sheaves of Lie algebroids L, provided they are equipped with a global

L-connection on L. Roughly speaking, we replace the local integrating groupoid by J(L), the

sheaf of L-jets associated to L. Then, using the L-connection, we define the exponential map

j∇ of the dual version of the PBW theorem:

Theorem. Let L be a locally free sheaf of Lie algebroids of constant rank r together with an

L-connection ∇L. There exists an OX-linear isomorphism of sheaves of algebras

j∇ : J(L) −→ SymOX(L∨)

with respect to the first OX-module structure on J(L).

For the definition of J(L) and its two OX-module structures we refer to §2.1.2. The algebra

structure on J(L) is induced by the coalgebra structure on U(L), thus it follows that the dual

map

j∨∇ : SymOX(L) −→ U(L), φ(j∨∇(X)) = j∇(φ)(X)

where X ∈ SymOX(L) and φ ∈ J(L), is an isomorphism of filtered coalgebras in the category of

OX-modules. Both the induced maps on the associated graded spaces are isomorphisms.

Now we briefly discuss the different subsections. To prove the PBW theorem we need parts

of the structures that are defined on U(L) and J(L), and we recollect them in §2.1. These

structures are described in more detail in [KP] and [CRvdB10]; the unversal enveloping U(L) is

a sheaf of so-called left Hopf algebroids, and J(L) is a sheaf of Hopf algebroids. Although not

formulated in this language, most of the structure maps of the (left) Hopf algebroids U(L) and

J(L) were already contained in [NeTs01]. In §2.2 we give the definition of the exponential map

j∇ and we prove the dual PBW theorem. We also give recursive relations for the definitions

29



30 CHAPTER 2. THE POINCARE–BIRKHOFF–WITT THEOREM

of j∇ and j∨∇. It turns out that the recursive relation that we find for j∨∇ coincides with the

recursive relation that was used in [LSX14] to prove the PBW theorem. Then we discuss the

straightforward extension of the PBW theorem to an isomorphism of differential operators on

J(L) and SymOX(L∨), which we use in chapter 4 and the appendix about formality for Lie

algebroids. Finally, we discuss how the PBW theorem depends on a linear combination of

connections and how it can be twisted by a locally free OX-module E.

Recently there has been an increased interest in the PBW theorem, see [LiS, LiSX, SX].

The application of the PBW theorem to the formality theorem that the authors consider is very

close to our appendix.

2.1 The universal enveloping algebra and the jet algebra

2.1.1 The universal enveloping algebra revisited

Let L be a locally free sheaf of Lie algebroids of constant, finite rank over a ringed space (X,OX).

The (sheaf of) universal enveloping algebra(s) U(L) has both a natural left and right OX-module

structure defined by left and right multiplication respectively. This leads to different notions of

tensor products, which we will denote as follows:

U(L)⊗ll U(L) :=(U(L)⊗K U(L))/ 〈rD⊗ E−D⊗ rD〉

U(L)⊗rl U(L) :=(U(L)⊗K U(L))/ 〈Dr⊗ E−D⊗ rD〉 .

This should make the general rule clear. In addition to the algebra structure on U(L), there

also exists a counital, cocommutative and coassociative OX-coalgebra structure on U(L). The

comultiplication ∆ : U(L) → U(L)⊗ll U(L) is defined by

∆(r) =r⊗ 1, r ∈ OX

∆(X) =X⊗ 1+ 1⊗ X, X ∈ L(2.1.1)

by an application of the universal property of U(L). The counit ε : U(L) → OX is defined by

ε(D) = D(1), D ∈ U(L).

Remark 2.1.1. The multiplication on U(L)⊗llU(L) by the obvious formula is not well-defined.

However, on the subspace of U(L)⊗ll U(L) which satisfies D⊗ E(f⊗ 1− 1⊗ f) = 0, the naive

product is well-defined. One can easily show that

∆(f)(g⊗ 1− 1⊗ g) = 0

∆(X)(g⊗ 1− 1⊗ g) = 0,

hence the map ∆ takes values in the subspace of U(L) ⊗ll U(L) where the multiplication is

defined, thus one can apply the universal property to obtain ∆ : U(L) → U(L) ⊗ll U(L). For

elements D,E ∈ U(L) it is given by ∆(DE) = ∆(D)∆(E) = D(1)E(1) ⊗D(2)E(2) using Sweedler

notation.

The anchor map ρ : L → Der(OX) extends, again by the universal property, to a map

ρ : U(L) → End(OX). In fact, the image of this map is contained in the differential operators

of OX, i.e. ρ : U(L) → Diff(OX), therefore the algebra U(L) is sometimes called the algebra of

L-differential operators. This map will be used in chapter 3. The universal enveloping algebra

admits a natural ascending filtration, denoted by

F0U(L) ⊂ F1U(L) ⊂ . . . , (2.1.2)



2.1. THE UNIVERSAL ENVELOPING ALGEBRA AND THE JET ALGEBRA 31

where elements of degree zero are given by i(OX) and U(L)p is defined to be the OX-submodule

of U(L) which is generated by elements of i(L)p. Because L is locally free, the associated

graded algebra is isomorphic to SymOX(L).

Remark 2.1.2. The universal enveloping algebra is a so-called left Hopf algebroid, c.f. [KK].

The notion of a left Hopf algebroid is weaker than the notion of a Hopf algebroid, as the authors

explained in [KP]; the difference being that left Hopf algebroids are not required to have an

antipode. Let us summarize the structures of the left Hopf algebroid U(L).

1. The universal enveloping algebra is endowed with a multiplication m. The unit is given

by the natural inclusion of OX in U(L).

2. The comultiplication

∆ : U(L) −→ U(L)⊗ll U(L)

which is defined in (2.1.1), is an OX-bimodule morphism, where both the left and right

module structures are given by left multiplication. The image of this map is contained

in the subspace on which the entrywise multiplication is defined. The counit for the

comultiplication is the composition of ρ : U(L) → End(OX) and the evaluation in 1:

ε : U(L) −→ OX, D → D(1).

It is an OX-bimodule morphism, where both the left and right module structures are given

by left multiplication.

3. The coproduct is multiplicative, i.e.

∆(DE) = ∆(D)∆(E).

4. The Hopf–Galois map:

β : U(L)⊗rlU(L) → U(L)⊗llU(L), D⊗ E → D(1) ⊗D(2)E

is an isomorphism.

2.1.2 The jet algebra

The structures

The jet algebra is simply defined as the dual of the universal enveloping algebra:

J(L) := HomOX(U(L),OX),

To be more precise, recall the ascending filtration

. . . ⊂ FkU(L) ⊂ Fk+1U(L) ⊂ . . . ,

on U(L). Define FkJ(L) := HomOX(FkU(L),OX), where the OX-linearity holds with respect to

the left OX-module structure on U(L). The jet algebra J(L) is defined as the projective limit:

J(L) = proj limk

FkJ(L).

The jet algebra is a Hopf algebroid, c.f. [CvdB, KP, NeTs01]. Let us discuss the structures of

this concept. We can dualize the cocommutative coproduct on U(L) to obtain a product on

J(L):

(φ1φ2)(D) := φ1(D(1))φ2(D(2)), φ1, φ2 ∈ J(L), D ∈ U(L). (2.1.3)




This defines a commutative algebra structure on J(L). The dual of the counit ε of U(L) is the

unit

ε : OX −→ J(L), 1 → (D → D(1)).

Analogous to the right and left OX-module structures on U(L), there exist two OX-module

structures on J(L), defined by

α1(r)(D) := ε(rD) = rε(D) = rD(1),

α2(r)(D) := ε(Dr) = D(r), r ∈ OX, D ∈ U(L).

Remark 2.1.3. We will use the different OX-module structures on J(L) frequently throughout

this thesis. When they occur in tensor products or homomorphisms, we will write this as

follows:

HomOXi(J(L),E) J(L)⊗OXi

E,

where E is an OX module. In the rare occasions that we consider the tensor product of J(L)

with itself, we will either explicitly mention which OX-module structures we use, or we will

write

J(L)αi⊗αj

J(L).

These OX-module structures can be upgraded to two U(L)-module structures. We formulate

this using the notion of a flat connection, since we have seen in (1.2.2) that U(L)-modules

corresponds to flat L-connections.

Lemma 2.1.4. There are two flat L-connections ∇(1) and ∇(2) on J(L) with respect to the

two OX-module structures α1 and α2. They are defined by

∇(1)X φ(D) := φ(XD) − X(φ(D)) (2.1.4a)

∇(2)X φ(D) := φ(DX) (2.1.4b)

for X ∈ L, φ ∈ J(L), D ∈ U(L), and act by derivations. As ∇(1) is compatible with α1

and ∇(2) with α2, the flat connections define two commuting U(L)-module structures on J(L),

written ·1 and ·2, which are explicitly given by

D ·1 φ(E) =D−(φ(D+)

D ·2 φ(E) =φ(ED)

Proof. In the literature, the connection ∇(1) is sometimes denoted by ∇G and referred to as

the Grothendieck connection. To prove that ∇(i) are connections, we have to show that

∇(i)rXφ(D) =αi(r)∇(i)

X φ(D) (2.1.5)

∇(i)X αi(r)φ(D) =αi(X(r))φ(D) + αi(r)∇(i)

X φ(D). (2.1.6)

First we give some useful identities:

α1(r)φ(D) =α1(r)(D(1))φ(D(2)) α2(r)φ(D) =α2(r)(D(1))φ(D(2))

=rα1(D(1))φ(D(2)) =(Dr)(1)(1)φ((Dr)(2))

=rφ(D) =φ(Dr).

For both identities we used that α1 is the unit for J(L), and on the RHS we also used ∆(r) = r⊗1.

Let us now prove Equation (2.1.5):

∇(1)rXφ(D) =r(∇(1)

X φ(D)) ∇(2)rXφ(D) =φ(DrX)

=α1(r)∇(1)X φ(D) =α2(r)∇(2)

X φ(D).



2.1. THE UNIVERSAL ENVELOPING ALGEBRA AND THE JET ALGEBRA 33

Before proving that Equation (2.1.6) holds, we show that ∇(i)X act by derivations. We use that

∆(XD) = XD(1) ⊗D(2) +D(1) ⊗ XD(2) and ∆(DX) = D(1)X⊗D(2) +D(1) ⊗D(2)X.

∇(1)X (φ1φ2)(D) =X(φ1φ2(D)) − φ1φ2(XD)

=X(φ1(D(1))φ2(D(2))) − φ1((XD)(1))φ2((XD)(2))

=X(φ1(D(1)))φ2(D(2)) + φ1(D(1))X(φ2(D(2))

− φ1(XD(1))φ2(D(2)) − φ1(D(1))φ2(XD(2))

=(∇(1)X φ1)φ2(D) + φ1(∇(1)

X φ2)(D)

and

∇(2)X (φ1φ2)(D) =φ1φ2(DX)

=φ1(D(1)X)φ2(D(2)) + φ1(D(1))φ2(D(2)X)

=(∇(2)X φ1)φ2(D) + φ1(∇(2)

X φ2)(D).

These properties, together with the easy fact that ∇(i)X αi(r) = αi(X(r)) for i = 1, 2, imply

Equation (2.1.6):

∇(i)X (αi(r)φ)(D) =∇(i)

X αi(r)φ(D) + αi(r)∇(i)X φ(D)

=αi(X(r))φ(D) + αi(r)∇(i)X φ(D).

The fact that both connections are flat is straightforward and is omitted. The expression for the

second U(L)-module structure is obvious, an the expression for the first U(L)-module structure

follows from the fact that it holds on degree 0 and 1 terms, and the fact that the action factorizes

through U(L)⊗rl U(L), together with the universal property. This finishes the proof.

We give a very easy, but important lemma that finds its application in the Fedosov resolu-

tions of chapter 4 and the appendix.

Lemma 2.1.5. The two U(L)-module structures on J(L) that were defined in the previous

lemma commute. In particular this implies that for all r ∈ OX the following holds:

∇(1)(α2(r)) = 0 ∇(2)(α1(r)) = 0.

Proof. Left to the reader.

The filtration

The ascending filtration on U(L) can be dualized to a descending filtration:

J(L) ⊃ J(L)1 ⊃ J(L)2 ⊃ . . . ,

where J(L)k : φ ∈ J(L) : φ(D) = 0 for all D ∈ Fk−1U(L). An alternative description of

this filtration is given by the following: The unit ε : J(L) → OX satisfies ker(ε) = J(L)1,

and a computation shows that (ker(ε))k = J(L)k. The inclusion to the right is fairly easy

if one uses that the coproduct on U(L) respects the filtration. For the left inclusion a local

argument suffices. We say that the descending filtration on J(L) is equal to the adic filtration

with respect to ker(ε). It is easy to see that the associated graded algebra to the filtration on

J(L) is isomorphic to SymOX(L∨). The adic filtration on J(L) defines the adic topology. The

open sets in this topology are given by J(L)k. One easily sees that the map

H : U(L) −→ HomcontOX1

(J(L),OX), D → (φ → φ(D))

is well-defined, and we have




Lemma 2.1.6 ([CRvdB10], lemma 5.1). The map H is an isomorphism of OX-bimodules.

Let us now summarize the Hopf algebroid structure on J(L) that is defined in [KP] and, in

a slightly different way, [CRvdB10]. We combine the two results:

Theorem 2.1.7 ([KP, CRvdB10]). The commutative, associative algebra J(L) is a Hopf alge-

broid, i.e. it has the following structure maps

1. A counit ε : J(L) → OX, defined by ε(φ) := φ(1), which is an algebra morphism of

OX-bimodules.

2. Two unit maps αi : OX → J(L) which are algebra morphisms. They satisfy εαi = id|OX.

3. A comultiplication ∆ : J(L) → J(L)α2⊗α1

J(L) which is a coassociative OX-bimodule

morphism, with the the left OX-module structure defined by α1, and the right OX-module

structure by α2, both on J(L) and J(L) → J(L)α2⊗α1

J(L).

4. An antipode S : J(L) → J(L) interchanging the two U(L)-module structures and satisfying

S2 = id|J(L).

The structure maps satisfy, for all φ ∈ J(L),

α1(ε(φ(1)))φ(2) =φ = φ(1)α2(ε(φ(2)))

α1 ε(φ) =S(φ(1))φ(2)

α2 ε(φ) =φ(1)S(φ(2)).

We omit the proof, but make a few remarks. In [KP] it is proved that the map φ ⊗ ψ →(D⊗ E → φ(Dψ(E))) is an isomorphism

J(L)α2⊗α1

J(L) ∼= lim←−p

Hom(Fp(U(L)⊗rl U(L)),OX).

Therefore the coproduct on J(L) can be defined by dualizing the product:

φ(DE) =: φ(1)(Dφ(2)(E))

where we used Sweedler notation. The OX-linearity of the multiplication on U(L) implies that

the comultiplication on J(L) is OX-bilinear. The antipode S is defined by

Sφ(D) := D·1φ(1)

and it is easy to check that it interchanges the two U(L)-module structures on J(L).

2.2 Incarnations and properties of the PBW theorem

2.2.1 The exponential map

In this section we construct a PBW map for the jet algebra of a Lie algebroid, depending on

the choice of a connection. Recall that Rinehart proved in [Rin] that gr(U(L, R)) ∼= SymR(L) for

a Lie–Rinehart pair (L, R) under the condition that L is projective over R. However, he did not

construct a map SymR(L) −→ U(L, R) inducing this isomorphism. Such a map is the content

of our PBW theorem. We fix a locally free sheaf of Lie algebroids L of constant rank r over a

ringed space (X,OX). We assume that the sheaf L admits an L-connection ∇L, and fix one.

Definition 2.2.1. Consider the sheaf L equipped with the zero bracket and the trivial action:

(L, [ , ] = 0, ρ = 0) over (X,OX). The symmetric algebra SymOX(L) is defined as the associ-

ated universal enveloping algebra of this algebroid. The associated jet algebra is given by

HomOX(SymOX

(L),OX) ∼= lim←−k

SymOX(L∨)/ SymOX

(L∨)k =: SymOX(L∨).



2.2. INCARNATIONS AND PROPERTIES OF THE PBW THEOREM 35

We denote the descending filtration on this jet algebra by SymOX(L) ⊃ J ⊃ J2 ⊃ . . .. The

L-connection ∇L on L induces L-connections on tensor products of copies of L and L∨ by the

usual formulas, which we all denote by ∇L. In particular, we obtain connections on the tensor

algebra TOX(L), the symmetric algebra SymOX

(L) and on SymOX(L∨).

The connection acts by derivations on the tensor algebra and the symmetric algebra. More-

over, the product on SymOX(L∨) is induced by the coproduct on SymOX

(L) and the connection

acts by coderivations with respect to this coproduct, as can be easily checked. Hence the connec-

tion acts by derivations on SymOX(L∨). The connection on SymOX

(L∨) can be symmetrized;

∇Ls β(X1 · · ·Xk) :=

1

k

k∑i=1

(∇LXiβ)(X1 · · · Xi · · ·Xk), β ∈ SymOX

(L∨), Xi ∈ L. (2.2.1)

We claim that this is a derivation of SymOX(L)∨ with the property

∇L : Jk −→ Jk+1. (2.2.2)

The latter fact is clear, and the derivation property follows from the fact that the connection

∇L acts by derivations. It is explained in more detail in the lemma below. Consider the algebra

J(L)⊗OX2SymOX

(L∨)

with entrywise multiplication. We define a connection D on this algebra by the formula:

DX(φ⊗ β) = ∇(2)X φ⊗ β+ φ⊗∇L

Xβ.

It is well-defined because

DX(α2(r)φ⊗ β) =α2(X(r))φ⊗ β+ α2(r)∇(2)X φ⊗ β+ α2(r)φ⊗∇L

Xβ

=φ⊗ X(r)β+∇(2)X φ⊗ rβ+ φ⊗ r∇L

Xβ

=DX(φ⊗ rβ)

and acts by derivations since both ∇(2) and ∇L both act by derivations. Let us remark that

the connection can be viewed as the natural connection on HomOX2(SymOX

(L), J(L)).

Lemma 2.2.2. The symmetrized version of this connection, denoted by Ds and defined by

Ds(φ⊗ β)(X1 · · ·Xk) :=

k∑i=1

DXi(φ⊗ β)(X1 · · · Xi · · ·Xk),

is a derivation with the property (2.2.2) for the component in SymOX(L∨).

Proof. We will compute the left and right hand side of Ds((φ ⊗ β1)(ψ ⊗ β1)) = (Ds(φ ⊗β1))(ψ⊗ β2) + (φ⊗ β1)Ds(ψ⊗ β2) evaluated on elements Xi ∈ L. Using that

β1β2(X1 · · ·Xk) :=∑σ∈Sk

β1(Xσ(1) · · ·Xσ(i))β2(Xσ(i+1) · · ·Xσ(k))

for β1, β2 ∈ SymOX(L∨) and the fact that D acts by derivations it follows that

Ds((φ⊗ β1)(ψ⊗ β2))(X1 · · ·Xk)

=

k∑i=1

DXi((φ⊗ β1)(ψ⊗ β2))(X1 · · · Xi · · ·Xk)

=∑σ∈Sk

((DXσ(1)

(φ⊗ β1))(Xσ(2) · · ·Xσ(i))(ψ⊗ β2)(Xσ(i+1) · · ·Xσ(k))

+ (φ⊗ β1)(Xσ(1) · · ·Xσ(i−1))(DXσ(i)(ψ⊗ β2))(Xσ(i+1) · · ·Xσ(k))

).




Moreover, we have

(Ds(φ⊗ β1))(ψ⊗ β2)(X1 · · ·Xk)

=∑σ∈Sk

(Ds(φ⊗ β1))(Xσ(1) · · ·Xσ(i))(ψ⊗ β2)(Xσ(i+1) · · ·Xσ(k))

=∑σ∈Sk

(DXσ(1)(φ⊗ β1))(Xσ(2) · · ·Xσ(i))(ψ⊗ β2)(Xσ(i+1) · · ·Xσ(k)),

and a similar expression for (φ⊗ β)(Dψ⊗ β). This proves the lemma.

The PBW map j∇ is now defined as the composition

J(L)−⊗1−→ J(L)⊗OX2

SymOX(L∨)

eD

−→ J(L)⊗OX2SymOX

(L∨)ev1−→ SymOX

(L∨), (2.2.3)

where ev1 means evaluation of a jet in 1 ∈ U(L). Note that this map is well-defined since the

operator Ds has property (2.2.2) and hence the exponent eD is well-defined. To be precise, we

pair the image of an element φ ∈ J(L) under the PBW morphism with X1 · · ·Xk ∈ SymOX(L):

jk∇(φ)(X1 · · ·Xk) := j∇(φ)(X1 · · ·Xk) :=1

k!Dk

s (φ⊗ 1)(1, X1 · · ·Xk) =:1

k!Dk

s (φ)(X1 · · ·Xk).

We introduced some notation along the way. We describe a recursive formula for this map in

the following proposition:

Proposition 2.2.3. For all k ∈ N the following identity holds:

jk∇(φ)(X1 · · ·Xk) =1

k

k−1∑i=1

∇LXi(jk−1

∇ (φ))(X1 · · · Xi · · ·Xk)

−1

k

∑i=1

jk−1∇ (∇(1)

Xiφ)(X1 · · · Xi · · ·Xk)).

(2.2.4)

Proof. This follows by a straightforward computation:

Dks (φ)(X1 · · ·Xk)(1) =

1

k

k∑i=1

∇(2)Xi

(Dk−1s φ(X1 · · · Xi · · ·Xk))(1)

−1

k

k∑i=1

Dk−1s (φ)(∇L

Xi(X1 · · · Xi · · ·Xk))(1)

=1

k

k∑i=1

Xi(Dk−1s φ(X1 · · · Xi · · ·Xk)(1))

−1

k

k∑i=1

Dk−1s φ(X1 · · · Xi · · ·Xk)(Xi)

−1

k

k∑i=1

Dk−1s (φ)(∇L

Xi(X1 · · · Xi · · ·Xk))(1),

where we see that the first and third part of the last expression combine to the first part of

Equation (2.2.4), and the middle part of the last expression is equal to the second part of

Equation (2.2.4).

Theorem 2.2.4. Let L be a locally free sheaf of Lie algebroids of constant rank r together with

an L-connection ∇L. The map

j∇ : J(L) −→ SymOX(L∨),




defined in (2.2.3), is an OX-linear isomorphism of sheaves of algebras with respect to the first

OX-module structure on J(L). The dual map

j∨∇ : SymOX(L) −→ U(L) φ(j∨∇(X)) = j∇(φ)(X),

where X ∈ Sym(L), φ ∈ J(L), is an isomorphism of filtered coalgebras in the category of

OX-modules. Furthermore, the induced map

gr(j∨∇) : SymOX(L) −→ gr(U(L))

is the canonical morphism defined considered by [Rin].

Proof. The OX-linearity of j∇ follows from the fact that the actions of ∇(2) and α1 commute

on J(L). Since the maps −⊗ 1, ev1 and the exponent of a derivation on an algebra are algebra

morphisms, j∇ is an algebra morphism. Recall the adic filtrations on J(L) and SymOX(L)

J(L) ⊃ I ⊃ I2 ⊃ . . . , SymOX(L) ⊃ J ⊃ J2 ⊃ . . .

Note that J(L) and SymOX(L) are complete with respect to those filtrations. Moreover, they

induce topologies on J(L) and SymOX(L) and in [CRvdB10, lemma 5.1] it is shown that

U(L) ∼= HomcontOX1

(J(L),OX), SymOX(L) ∼= Homcont

OX(SymOX

(L∨),OX).

Explicitly, it means that D ∈ HomcontOX1

(J(L),OX) if and only if there exists an n such that

D(φ) = 0 for all φ ∈ In. Now we show that j∇ respects these filtrations. Let φ ∈ J(L). Using

Proposition 2.2.3 it is easy to prove the euqation

j∇(φ)(X1 · · ·Xk) =

=1

k!

∑

σ∈Sk

φ(Xσ(1) · · ·Xσ(k)) −∑σ∈Sk

φ(∇LXσ(1)

(Xσ(2) · · ·Xσ(k))) + φ(X)

,

(2.2.5)

where X ∈ Fk−2U(L) is a complicated term involving the connection. In the remark after this

proof we give a more explicit inductive formula from which the previous one can be deduced.

Hence, if φ ∈ Ik, i.e. φ(D) = 0 for all D ∈ Fk−1U(L),

j∇(φ)(X1 · · ·Xm) = 0, forall m < k,

i.e. j∇(φ) ∈ Jl. Because j∇ respects the filtrations, the dual map j∨∇ is well-defined. Moreover,

j∨∇ respects the coalgebra structures because j∇ respects the product structures and it is OX-

linear because j∇ is. This leaves us to show that j∇ and its dual are isomorphisms, which follows

once we show that the associated graded morphism gr(j∨∇) is an isomorphism. According to

[Rin], the canonical morphism

SymOX(L) −→ gr(U(L)), X1 · · ·Xk → [i(X1) · · · i(Xk)]

is an isomorphism, hence if we show that gr(j∨∇) is the canonical morphism, the required state-

ments follow. Equation (2.2.5) shows that the morphism j∨∇ is given by

j∨∇(X1 · · ·Xk) =1

k!

∑

σ∈Sk

(Xσ(1) · · ·Xσ(k) −∇L

Xσ(1)(Xσ(2) · · ·Xσ(k))

)+ X

,

and this shows that gr(j∨∇) is indeed equal to the canonical morphism.

Example 13. The adjoint representation ad of a Lie algebra g is a g-connection on g. For

elements X1 · · ·Xk ∈ Symk(g), the PBW map is inductively defined by

j∨ad(X1 · · ·Xk) :=1

k

k∑i=1

(Xi · j∨ad(X1 · · · Xi · · ·Xk) − j∨ad(ad(Xi)(X1 · · · Xi · · ·Xk))

),




see Equation (2.2.6). We claim that the second term vanishes for each k. This follows easily from

the observation that each term X1 · · · [Xi, Xj] · · ·Xk is cancelled by the term X1 · · · [Xj, Xi] · · ·Xk

because the connection is symmetrized. The PBW map Sym(g) → U(g) induced by the adjoint

representation is therefore equal to the symmetrization map.

2.2.2 Compatibility with the smooth approach

Our approach to the PBW theorem via the dual map defined on jets can be related to that of

[LSX14]. There, the authors work in the smooth category and consider pairs of Lie algebroids,

which we will define below. First we relate our definition of the dual PBWmap to their recursive

definition of the PBW map. Consider X := X1 · · ·Xk ∈ Symk(L) and φ ∈ J(L). Then

φ(j∨∇(X)) =⟨j∨∇(φ), X

⟩

=1

k!Dkφ(X)(1)

=1

k!

k∑i=1

∇(2)Xi

(Dk−1φ(X1 · · · Xi · · ·Xk))(1) −Dk−1φ(∇LXi(X1 · · · Xi · · ·Xk))

=1

k

k∑i=1

(φ(Xi · j∨∇(X1 · · · Xi · · ·Xk)) − φ(j∨∇∇L

Xi(X1 · · · Xi · · ·Xk))

).

Therefore, we find the recursive formula

j∨∇(X1 · · ·Xk) =1

k

k∑i=1

(Xi · j∨∇(X1 · · · Xi · · ·Xk)) − j∨∇(∇Xi

(X1 · · · Xi · · ·Xk))). (2.2.6)

This is precisely the definition of the PBW morphism in [LSX14, Remark 2.3] for the inclusion

of the trivial Lie algebroid into L.

For the remaining part of this subsection we work in the smooth category. As mentioned in

the introduction of this chapter, in [NWX] the authors use the concept of a local Lie groupoid

to define an exponential map in a geometrical way. The fact that each smooth Lie algbroid

admits a local integration implies that this construction works for all smooth Lie algebroids. In

[LSX14], the authors showed that their algebraic recursive definition for a map

Sym(A) −→ U(A)

coincides with the exponential map that is provided by the local Lie groupoid of [NWX]. We

shortly discuss the geometrical construction from [NWX] and [LSX14]. To be precise, we work

in the setting of [LSX14] and we use their notations for this subsection. Their initial data

consists of an exact sequence of smooth, integrable Lie algebroids

0 −→ B −→ A −→ A/B −→ 0,

together with a splitting j : A/B −→ A and an A-connection on A/B which extends the Bott

connection. The Bott connection is the B connection on A/B defined by

∇Botta q(a) := q([b, a]) a ∈ A, b ∈ B.

Standard examples of such exact sequences come from holomorphic Lie algebroids and foliations.

They first observe that, given a vector bundle E → M, an A-connection on E can be defined as

a horizontal lifting h : L×M E → TE using the formula:

h(am, em) := e∗(ρ(am)) − τem((∇ae)m).




In the above formula, m ∈ M, a and e are extensions of am ∈ Am and em ∈ Em, e∗ is the

pushforward along e, and τemis the identification of Em with its tangent space at the point

em. This interpretation gives rise to the geodesic vector field Ξ on A/B, defined by

Ξx := h(j(x), x), x ∈ A/B (2.2.7)

where h : A×M A/B → T(A/B) is the horizontal lifting of the A-connection on A/B extending

the Bott connection. Before we describe how the geodesic vector field is used to define the

exponential map, let us relate it to the degree one derivation on Sym(A∨) that we defined to

construct the PBW map. The formula for the derivation on Sym(A∨) can be generalized to a

formula for a derivation on Sym((A/B)∨):

D(β)(x1 · · · xn) :=n∑

i=1

∇j(xi)β(x1 · · · xi · · · xn) (2.2.8)

for xi ∈ A/B and β ∈ Sym((A/B∨). Note that Sym((A/B)∨) maps injectively into C∞(A/B)

by the standard formula

ι(β)(xm) :=

⟨β, xm · · · xm︸︷︷︸

n times

⟩

for β ∈ Symn((A/B)∨) and xm ∈ (A/B)m. For degree 0 elements it is given by

ι = π∗ : C∞(M) → C∞(A/B).

The completion Sym((A/B)∨) however, does not have this property, hence we can only relate

our derivation to the geodesic vector field on the subalgebra of polynomial functions. The

following lemma was suggested by Ping Xu.

Lemma 2.2.5. On the subalgebra Sym((A/B)∨) ⊂ C∞((A/B), the geodesic vector field as

defined in (2.2.7) is equal to the derivation defined in (2.2.8).

Proof. Since both the geodesic vector field and the derivation act by derivations, and since

Sym((A/B)∨) is generated by degree 0 and 1 elements, we only have to check the claim for

these degrees. Let π∗(f) ∈ C∞(A/B), where π : A/B → M denotes the projection. We have to

prove that Ξxm(π∗(f))(xm) = ι(D(f))(xm), where xm ∈ A/B. We compute the action of the

geodesic vector field:

Ξxm(π∗(f))(xm) :=x∗(ρ(j(xm))(π∗(f))(xm) − τxm

(∇j(x)x)(π∗(f))(xm)

=ρ(j(xm))(π∗(f) x)(m)

=ρ(j(xm))(f)(m)

=π∗(ρ(j(xm))(f))(xm)

where we used that the second term on the RHS involves a vector tangent to the fiber (A/B)m,

which annihilates π∗(C∞(M)), and where x stands for a section of A/B extending xm. On the

other hand, Equation 2.2.8 immediately gives that

ι(D(f))(xm) = 〈D(f), xm〉 (m)

=ρ(j(xm))(f)(m)

=π∗(ρ(j(xm)(f))(xm).

Now we turn to de degree 1 case. Let β ∈ Sym1((A/B)∨). We have to prove that Ξxm(ι(β))(xm) =

ι(D(β))(xm), which is shown by the following computation:

Ξxm(ι(β))(xm) =x∗(ρ(j(xm))(ι(β)))(xm) − τxm

(∇j(x)x)(ι(β))(xm)

=ρ(j(xm))(ι(β) x)(m) −⟨β,∇j(x)x

⟩(m)

=ρ(j(xm))(〈β, xm〉)(m) −⟨β,∇j(x)x

⟩(m)

= 〈D(β), xm ⊗ xm〉 = ι(D(β))(xm).




Hence, the lemma is proved.

Now we briefly discuss the exponential map using integrations. Let A be a Lie groupoid

integrating A and B be a Lie groupoid integrating B.

Definition 2.2.6. Let A be a Lie groupoid integrating a Lie algebroid A.

1. An s-path is a smooth curve γ : I → A emanating from a point in M ⊂ A and fully

contained in an s-fiber.

2. A ρ-path is a smooth curve β : I → A satisfying

ρ(β(t)) = π∗(β′(t)).

Every ρ-path β uniquely determines an s-path γ and vice versa, by the following formula:

β(t) =d

dτ(γ−1(t)γ(τ))t.

Now we describe the exponential map as defined in [NWX, LSX14]. Let x be an element in A/B

in the neighborhood of the zero section. Consider the integral curve βx(t) emanating from x

defined by the geodesic vector field. The composition j(βx(t)) is a ρ-path in B, hence it lifts to an

s-path gx(t) in A . The composition of this path with the canonical projection p : A → A /B,

and finally the evaluation in t = 1, defines the exponential map, i.e. exp∇,j(x) := p(gx(1)).

It is proved in [LSX14] that it induces a local diffeomorphism A/B → A /B around the zero

sections.

Given a smooth submersion p : P → M together with a section ε, consider the space

D(P,M), which are maps C∞(P) → C∞(M) given by the composition of a differential operator

on P tangent to the fiber, and ε∗.

To relate the exponential map to the PBWmap Sym(A/B) → U(A)/U(A)Γ(B), the following

two identifications are used:

• The algebra Γ(Sym(A/B)) is isomorphic to the algebra D(A/B,M).

• The algebra U(A)/U(A)Γ(B) is isomorphic to the algebra D(A /B,M) with respect to

the smooth submersion s : A /B → M.

Theorem 2.2.7 ([LSX14]). The local diffeomorphism of fiber bundles

exp∇,j : A/B → A /B

induces an isomorphism of coalgebras D(A/B,M) → D(A /B,M). This isomorphism coincides

with the isomorphism that is defined by the recursion (2.2.6).

2.2.3 The noncommutative PBW theorem

In this subsection we describe a noncommutative version of the PBW theorem for Lie algebroids.

It is also contained in the appendix, but we include it here because it is used in chapter 4. Given

a locally free sheaf of Lie algebroids of constant rank r, the inclusion α2 : OX → J(L) can be

extended to inclusions

L −→ J(L)⊗OX2L, X → α2(1)⊗ X

U(L) −→ J(L)⊗OX2U(L), D → α2(1)⊗D.




Lemma 2.2.8 ([CvdB], Lemma 4.3.2). Given a locally free Lie algebroid over OX, one has the

following isomorphisms:

J(L)⊗OX2L ∼= DerOX1

(J(L)) φ⊗ X → (ψ → φ∇(2)ψ)

J(L)⊗OX2U(L) ∼= DiffOX1

(J(L)) φ⊗D → (ψ → φD·2ψ)

where D·2 indicates the U(L)-module structure (2.1.4b) on J(L). The first is a J(L)-module

isomorphism of Lie algebras, and the second is a J(L)-linear isomorphism of algebras.

It is proved by studying the associated graded spaces with respect to the filtrations on U(L)

and J(L). The composition of differential operators in DiffOX1(J(L)), which is the algebra

structure, corresponds, under the isomorphism above, to the product

(φ⊗D) · (ψ⊗ E) := φD(1)(ψ)⊗D(2)E (2.2.9)

on J(L)⊗OX1U(L). The PBW map, defined in 2.2.4, is an OX-linear isomorphism

J(L) ∼= SymOX(L∨)

with respect to the first OX-module structure on J(L), hence the map j∇(K)(β) = j∇(K(j−1∇ (β)))

induces an algebra isomorphism

j∇ : DiffOX1(J(L))

∼=−→ DiffOX(SymOX

(L∨)

for K ∈ DiffOX1(J(L)) and β ∈ SymOX

(L∨). This proves the following theorem.

Theorem 2.2.9. Let L be a locally free Lie algebroid of constant, finite rank over a ringed

space (X,OX). Any L-connection ∇ on L induces an isomorphism of sheaves of algebras

j∇ : J(L)⊗OX2U(L)

∼=−→ DiffOX(SymOX

(L∨))

2.2.4 The derivative of the PBW map

In subsection we study the time-dependence of the PBW map on the family of connections

∇t = ∇1 + tγ, t ∈ [0, 1].

where γ ∈ Ω1L(End(L)). Clearly, this family of connections induces a family

jt := j∇t: J(L) −→ SymOX

(L∨)

of PBW isomorphisms. We define θt via the equation:

d

dt(jt(φ)) = θt(jt(φ)).

One has the following isomorphism for the OX-linear derivations on SymOX(L∨):

DerOX(SymOX

(L∨) ∼= SymOX(L∨)⊗ L.

Derivations X ∈ DerOX(SymOX

(L∨) decompose as

X =

∞∑i=−1

Xi

where Xi raises the degree of elements β ∈ SymOX(L∨) by i.




Proposition 2.2.10. The map θt is an OX-linear derivation of SymOX(L∨). If we write

θt =∑∞

k=−1 θit as above, the formulas

θ−1t = 0 = θ0t , and θ1t = γs

hold for all t ∈ [0, 1].

Proof. The fact that θt is an OX-linear derivation follows from the fact that jt is an algebra

morphism which is OX-linear. Now recall that jt = ev1 exp(Dst) −⊗ 1, where

Dst :=

(∇(2) ⊗ 1+ 1⊗∇t

)s

is a derivation of degree 1 on J(L)⊗OX1SymOX

(L∨) with respect to the degree in SymOX(L∨).

The derivative of exp(Dst) is given by the following general formula, which follows from a

straightforward computation:

d

dt(expDs

t) =exp adDs

t− 1

adDst

(dDs

t

dt

) expDs

t. (2.2.10)

Note that Dst and dDs

t/dt do not commute in general. It is easy to see that the derivative dDst/

dt = id⊗γs, where γs is the OX-linear derivation of degree 1 on SymOX(L∨) that corresponds

to γ ∈ Ω1L(End(L)) = (L∨)⊗2 ⊗L via the natural projection (L∨)⊗2 ⊗L → Sym

2

OX(L∨)⊗L.

Hence, since Dst has degree 1, γs is the term with the lowest degree in θ, and the statement

follows since the evaluation map ev1 intertwines id⊗ γs and γs.

2.2.5 Twisting by a locally free sheaf

In the final section on the PBW theorem, we discuss how to incorporate a locally free sheaf E

of OX-modules of finite, constant rank into the theory. Following [CvdB], we define the sheaf

of L-jets of E of order k as

Jk(L;E) := Jk(L)⊗OX2E.

We denote the full sheaf of infinite jets of E, which is a projective limit, by J(L;E), and define

the universal enveloping algebra with coefficients in E as

U(L;E) := HomcontOX1

(J(L;E),E) .

The following lemma follows directly from the definitions.

Lemma 2.2.11. The sheaf J(L;E) of L-jets of E has the structures of:

1. An OX1-linear J(L)-module by the assignment (φ,ψ⊗ s) → φψ⊗ s.

2. An OX1-linear J(L)-comodule by the assignment φ⊗ s → φ(1) ⊗ φ(2) ⊗ s.

Remark 2.2.12. The first oder jets fit into a short exact sequence

0 → L∨ ⊗ E → J1(L;E) → E → 0,

and a splitting σ : E → J1(L;E) of this sequence in the category of sheaves of OX1-modules is

equivalent to an L-connection on E.

Lemma 2.2.13. The universal enveloping algebra with coefficients in L admits a product by

the following formula:

D E(φ⊗ s) = D(φ(1) ⊗ E(φ(2) ⊗ s)), φ⊗ s ∈ J(L;E), D, E ∈ U(L;E)

where φ(1) ⊗ φ(2) = ∆(φ) is the coproduct on J(L).




Proof. The fact that D E is OX1-linear follows from the the fact that the coproduct

∆ : J(L) −→ J(L)⊗ J(L)

is OX1-linear with respect to the OX1

-module structure on the left copy of J(L) on the RHS.

One should think about U(L;E) as L-valued differential operators acting on E, with the

product simply given by composition. In fact, there is an equivalent definition of U(L;E) which

is based on the Atiyah algebroid which is closer to this interpretation. It seems however easier

to generalize the PBW theorem to the twisted case via the approach above. Recall that the

Atiyah algebroid AtL(E) of E with values in E fits into the following diagram

0 End(E) AtL(E) L 0

with exact rows and AtL(E) defined as in Example 1.1.1. One can also define AtL(E) as the

subsheaf EndK(E) of elements Y such that

[Y, f] = σY(f) f ∈ OX,

for σY ∈ L. It is easy to check that the commutator and the morphism Y → σY together with

the anchor map endow AtL(E) with the structure of a Lie algebroid.

Lemma 2.2.14. Given a Lie algebroid L over (X,OX) and a locally free OX-module E, we have

the following canonical isomorphism:

U(AtL(E)) ∼= U(L;E).

Proof. The proof is based on the universal property for U(AtL(E)). We define maps

OX −→ U(L;E) f → (f : φ⊗ s → fφ(1)s)

AtL(E)) −→ U(L;E) Y → (Y : φ⊗ s → φ(1)Y(s) + φ(σY)s).

It is easy to see that f ∈ U(L;E) and that Y is OX1-linear. Moreover, we have

Y(α2(f)φ⊗ s) = fφ(1)Y(s) + φ(σYf)s

= fφ(1)Y(s) + φ(1)σY(f)s+ fφ(σX)s

= φ(1)Y(fs) + φ(fσX)s

= Y(φ⊗ fs),

thus Y ∈ U(L;E). To apply the universal property, we have to show that

fY = f Y, σY(f) = [Y, f], [Y, X] = [Y, X].

The first equality is a straightforward check. We compute:

Y f(φ⊗ s) = Y(φ(1) ⊗ φ(2)(1)fs)

= φ(1)(1)Y(φ(2)(1)fs) + φ(1)(σY)φ(2)(1)fs

=(φ(1)(1)σY(φ(2)(1)) + φ(1)(σY)φ(2)(1)

)fs

+ φ(1)(1)φ(2)(1))fY(s) + φ(1)(1)φ(2)(1))σY(f)s

= φ(σY)fs+ φ(1)Y(s) + φ(1)σY(f)s

= f Y(φ⊗ s) + σY(f)(φ⊗ s),

where we used the definition φ(1)(Dφ(2)(E) = φ(DE) for the coproduct where D, E ∈ U(L) in

the fourth equality. This implies the second equation. For the third equation we compute:

Y X(φ⊗ s) = Y(φ(1) ⊗ (φ(2)(1)X(s) + φ(2)(σX)s))

= φ(1)(1)(σY(φ(2)(1))X(s) + φ(2)(1)YX(s) + σY(φ(2)(σX))s+ φ(2)(σX)Y(s)

)

+ φ(1)(σY)(φ(2)(1)X(s) + φ(2)(σX)s

).




The terms in the commutator [Y, X] which end with s are given by

φ(1)(1)σY(φ(2)(σX))s+ φ(1)(σY)φ(2)(σX)s

− φ(1)(1)σX(φ(2)(σY))s− φ(1)(σX)φ(2)(σY)s

= φ(σYσX − σXσY)

where we applied the rule

φ(σXσY) =φ(1)(σXφ(2)(σY))

=φ(1)(σX(φ(2)(σY))) + φ(1)σX)φ(2)(σY).

The terms ending with Y(s) and X(s) vanish as can be shown by a similar computation, and

the terms ending with YX(s) and XY(s) are

φ(1)YX(s) − φ(1)XY(s).

Since the commutator [Y, X] in the Atiyah algebroid has σ[Y,X] = [σY , σX] as associated elements

in L, this proves the third equality. The universal property implies that there exists an algebra

homomorphism

U(AtL(E)) −→ U(L;E).

The RHS has a natural ascending filtration defined by the descending filtration on J(L), and

the LHS has a filtration which is defined by defining

D ∈ FkU(L;E) if D = X1 · · ·XiYi+1 · · · Yl, i k

where σYj= 0 for all j. It is easy to see that the map on the associated graded spaces is an

isomorphism .

Lemma 2.2.15. Let ∇E be a flat L-connection on E. Then there exists an algebra morphism:

a∇E : U(L) −→ U(L,E).

Proof. Since a flat connection is given by a morphism of Lie algebras

∇E : L −→ AtL(E)

which is compatible with the OX-module structures and the action on OX, this follows from the

universal property.

Theorem 2.2.16. Let L be a Lie algebroid over a ringed space (X,OX), and E a locally free

OX-module of constant finite rank. Any pair (∇L,∇E) of L-connections on L and E induces

an OX1-linear isomorphism

j∇L,∇E : J(L;E) → SymOX(L∨)⊗OX

E

such that

j∇L,∇E(φψ⊗ e) = j∇L(φ)j∇L,∇E(ψ⊗ e) (2.2.11)

holds for any φ ∈ J(L), ψ⊗ e ∈ J(L;E).

Proof. The connections ∇L,∇E,∇(2) define a connection on SymOX(L∨) ⊗OX2

J(L;E) and

symetrizing this connection leads to a map of degree 1

Ds : SymOX(L∨)⊗OX2

J(L;E) → SymOX(L∨)⊗OX2

J(L;E)

which has the following Leibniz property:

Ds(rs) = Ds(r)s+ rDs(s), r ∈ SymOX(L∨)⊗OX2

J(L), s ∈ SymOX(L∨)⊗OX2

J(L;E).




Here Ds(r) stands for the derivation on J(L)⊗OX2SymOX

(L∨) defined in (2.2.4) acting on r,

and Ds(rs) and Ds(s) stands for the map that we just defined. Moreover, we used the natural

SymOX(L∨) ⊗OX2

J(L)-module structure on SymOX(L∨) ⊗OX2

J(L;E). The PBW map with

coefficients is now defined as the composition

J(L;E)1⊗−−→ SymOX

(L∨)⊗OX2J(L;E)

eD

−→ SymOX(L∨)⊗OX2

J(L;E)ev1−→ SymOX

(L∨)⊗ E

The Leibniz property of Ds gives property (2.2.11). The OX1-linearity and the fact that it is

an isomorphism follows in the same way as in the proof of Theorem 2.2.4.

Corollary 2.2.17. Using the notation from the previous theorem, any pair (∇L,∇E) of L-

connections on L and E induces an isomorphism

j∗∇L,∇E : SymOX(L)⊗OX

End(E) → U(L;E)

Proof. This follows from Sym(L)⊗OXEnd(E) ∼= Homcont

OX(SymOX

(L∨)⊗E,E) and the fact that

j∇L,∇E is OX1-linear.

There is also a noncommutative, twisted PBW theorem, which we describe now.

Theorem 2.2.18. The data of L-connections ∇L and ∇E on L and E induces an isomorphism

of sheaves of algebras:

J(L)⊗OX2U(L;E) −→ DiffOX

(SymOX(L∨)⊗OX

E).

Proof. Similar to the untwisted case, there is a canonical morphism

J(L)⊗OX2U(L;E) → DiffOX1

(J(L;E))

(φ⊗ e⊗D⊗ f) → (ψ⊗ s → φD ·2 (α2(f(s))ψ)⊗ e).

Where e, s ∈ E, f ∈ E∨, φ,ψ ∈ J(L), D ∈ U(L) and D acts on J(L) using ∇(2). Recall

that differential operators of order k on the J(L)-module J(L;E) are endomorphisms of J(L;E)

satisfying [φ0, [. . . , [φk, d] . . .] = 0 where φi ∈ J(L) are considered as endomorphisms of J(L;E).

It is straightforward to check that the morphism above is well-defined and has the range that is

indicated. Moreover, it respects the filtrations from U(L) and J(L) on the LHS and the degree

of differential operators and the J(L) filtration on the RHS. The fact that it is an isomorphism

follows from the same argument as in the untwisted case. Using now that Diff(Sym(L∨)⊗E) ∼=

DiffOX1(J(L;E)) by the morphism D → j∇L,∇E D j−1

∇L,∇E , the claim follows.






Chapter 3

Cyclic theory of the universal

enveloping algebra

This chapter can be divided into two parts. In the first, we relate the linear Poisson geometry

of A∨, the dual of a smooth Lie algebroid A → M, to the symmetric powers of the adjoint

representation (up to homotopy) of the Lie algebroid A. The result is an identification between

the polynomial Poisson cohomology complex and the complex of the representation. Moreover,

we show that the Poisson homology complex of A∨ can be related the tensor product of the

symmetric powers of the adjoint representation and the natural representation QA -the bundle

of A-densities.

In the second part we compute the Hochschild and cyclic (co)homology of the universal

enveloping algebra U(A) of A. First we define the Hochschild and cyclic (co)homology taking

into account the natural topology on U(A). The natural filtration on the universal enveloping

algebra leads to a spectral sequence that, in principle, allows one to compute the Hochschild

and cyclic (co)homology. However, it is not a priori clear that it degenerates. We solve this

problem by constructing maps on the level of chains from the polynomial Poisson (co)homology

complex to the Hochschild (co)complex of U(A). Finally, we discuss the relation between our

results and the existing literature. More extensive introductions can be found at the beginning

of the two sections.

3.1 Poisson (co)homology and representations up to ho-

motopy

In this section we consider the Poisson geometry of A∨, the dual of a smooth Lie algebroid

A → M. Indeed, it is well-known that Lie algebroid structures on a given vector bundle A → M

are in one-to-one correspondence with Poisson structures on A∨ which are linear along the

fibers, see [CM, prop. 7] for a proof of this fact. Here we shall prove that his observation can be

extended to a correspondence between (symmetric powers of) the adjoint representation (up to

homotopy) of the Lie algebroid A, and the linear (polynomial) Poisson cohomology complex of

the Poisson structure on A∨. Moreover, the Poisson homology complex of the Poisson structure

on A∨ can be related to yet another representation up to homotopy; the tensor product of the

symmetric powers of the adjoint representation and the natural representation QA -the bundle

of A-densities.

47



48 CHAPTER 3. CYCLIC THEORY OF THE UNIVERSAL ENVELOPING ALGEBRA

3.1.1 Polynomial polyvector fields and differential forms

First we assume A → M to be a smooth vector bundle. Any section s ∈ Γ(A) induces a fiberwise

linear function s on A∨ by the formula

s(α) :=⟨sπ(α), απ(α)

⟩π(α)

, α ∈ A∨,

where 〈 , 〉x denotes the natural pairing between Ax and its dual A∨x , for any x ∈ M. Sections

of the symmetric algebra bundle SymA likewise induce fiberwise polynomial functions on A∨

by the formula g(α) := s1 · · · sk(α) :=< s1, α > · · · < sk, α >, where g = s1 · · · sk ∈ Γ(SymA)

is the symmetric product of sections si ∈ Γ(A). We will sometimes write:

C∞lin(A

∨) = Γ(A), C∞pol(A

∨) = Γ(SymA).

It is clear that the functions on A∨ which are polynomial along the fibers form a graded algebra,

and the degree of elements can be detected as follows. The R>0-action by scaling A along the

fibers of the projection to the base M is generated by the Euler vector field E, which, choosing

a local frame ei for A, can be expressed as

E =∑

ei∂

∂ei. (3.1.1)

It acts on C∞(A) using the Lie derivative and detects the degree of a monomial, i.e., LE(g) = kg,

for g ∈ Γ(M, Symk A).

Although we can identify the sections of A or SymA with linear, respectively polynomial

functions along fibers on A∨, we prefer to only use this observation as inspiration, and to work

with the bundles themselves as well as their sections.

In this way, we can view the pair (M, SymA) as a locally ringed space. Regarding the

notation, we will not distinguish between the vector bundle SymA and its sheaf of sections.

The R-linear derivations of this sheaf

DerR(SymA) =: L

form a SymA-module. It is the space of sections of a vector bundle Der(SymA) over M, as the

following lemma shows. It is analogous to [L-BM, prop. 4.1].

Proposition 3.1.1. There is a short exact sequence of sheaves of Lie algebroids over (M, SymA):

0 −→ SymA⊗A∨ −→ DerR(SymA) −→ SymA⊗ TM −→ 0. (3.1.2)

A choice of a splitting of the global sections of this exact sequence is equivalent to a choice of a

TM-connection on A. In particular, DerR(SymA) is given by the sheaf of sections of a vector

bundle Der(SymA) over M.

Proof. First we describe the maps in the sequence above. The inclusion SymA ⊗ A∨ →DerR(SymA) is defined by the contraction ια : Symk A → Symk−1 A for α ∈ A∨. The map

DerR(SymA)⊗T∗M → SymA given by D⊗df → D(f) defines the projection DerR(SymA) −→SymA⊗TM. These maps are SymA-linear and their composition is equal to 0. The injectivity

of the first map is clear, whereas the surjectivity of the second follows from the fact that

a choice of a connection ∇ on A defines a SymA-linear splitting by g ⊗ X → g∇X where

g⊗X ∈ SymA⊗TM. In particular, this splitting gives an isomorphism of the C∞(M)-modules

DerR(Γ(SymA)) ∼= Γ(SymA⊗ (A∨ ⊕ TM), and hence DerR(Γ(SymA)) is given by the sheaf of

sections of a vector bundle Der(SymA).

Hereafter we will use the notation L for DerR(SymA). The grading on SymA induces a

grading on L. Let us describe the exact sequence of the lemma above in low degrees. It is clear



3.1. POISSON (CO)HOMOLOGY AND REPRESENTATIONS UP TO HOMOTOPY 49

that L<−1 is zero. In degree −1 it only consists of two copies of A∨, and in degree 0 it is given

by the short exact sequence

0 −→ A⊗A∨ −→ L0 −→ TM −→ 0

where the middle bundle is the Atiyah algebroid, see equation (1.1.1), which is often used to

define connections as splittings of this sequence.

Recall from proposition 1.1.12 that the Lie algebroid structure on L defines the Schouten–

Nijenhuis bracket [·, ·]SN, the de Rham differential dL and the Lie module structure L on the

L-polyvectorfields and the L-differential forms:

(TLpoly(A

∨),∧, [·, ·]SN) (

ΩL(A∨),∧, dL

).

L (3.1.3)

The notation is slightly inconsistent here; the polyvector fields associated to the sheaf of Lie

algebroids L over (M, SymA) should be denoted by TLpoly(M) if we would follow the notation

from the last chapter, but we chose TLpoly(A

∨) to indicate that the structure sheaf is SymA,

the polynomial functions on A∨. A similar remark holds for ΩL(A∨). Note that we view the

objects as sheaves over M. The isomorphism L ∼= SymA⊗ (A∨ ⊕ TM) from proposition 3.1.1

can be extended to the polyvector fields TLpoly(A

∨) and the dual of the exact sequence (3.1.2)

gives a similar isomorphism for the L-de Rham forms:

TLpoly(A

∨)∇∼=⊕k

⊕i+j=k

SymA⊗ (

i∧A∨ ⊗

j∧TM) (3.1.4)

ΩL(A∨)

∇∼=⊕k

⊕i+j=k

SymA⊗ (

i∧A⊗

j∧T∗M), (3.1.5)

both of which are SymA-linear. Perhaps this a good point to stipulate that there are two

natural gradings on these algebras; the polynomial one which is given by assigning

deg(A) = 1 deg(TM) = 0

deg(A∨) = −1 deg(T∗M) = 0

and the ”cohomological” degree which is given by i+ j in equations (3.1.4) and (3.1.5).

It is easy to see that the graded commutative products of the LHS correspond to the products

on the RHS which are given by the natural symmetric product on SymA and the wedge product

on the graded commuting parts. The insertion operator ι of polyvector fields into differential

forms is given by the natural pairings of A and A∨, TM and T∗M, and the multiplication on

SymA. The following lemma describes the other natural structures on the polyvector fields

and differential forms of L on the RHS of equations (3.1.4) and (3.1.5)

Lemma 3.1.2. Under the identifications (3.1.4) and (3.1.5), we have the following:

1. The Schouten–Nijenhuis bracket is determined by the formula

[(α,X), (β, Y)] = ∇Xβ−∇Yα+ [X, Y] + R∇(X, Y),

where X, Y ∈ ∧1TM and α,β ∈ ∧1A∨, R∇ is the curvature and R∇(X, Y) ∈ Sym1 A ⊗∧1A∨.

2. The de Rham differential dA∨ on ΩL(A∨) corresponds to d∇, which, on polynomial

functions, splits as

Symk A Symk−1 A⊗Λ1A

Symk A⊗Λ1T∗M

δ

∇




where δ is the Koszul differential on SymA. On differential forms it is detemined by the

formulas

dM : ΛkT∗M −→ Λk+1T∗M

and on ∧A it splits as

ΛkA ΛkA⊗Λ1T∗M

Sym1 A⊗Λk−1A⊗Λ2T∗M.

∇

R∇

Together with the compatibility with the wedge product this determines the de Rham dif-

ferential.

3. The Lie module structure L is determined by the following formulas:

LX(s) =∇Xs− R∇(X, s) Lα(s) = ∇α(s)

LX(ξ) =LMX ξ Lα(ξ) = 0,

where X ∈ ∧1TM, α ∈ ∧1A∨, s ∈ ∧1A and ξ ∈ ∧1T∗M.

Proof. The Schouten–Nijenhuis bracket is determined, using Equation (1.1.3), by the bracket

evaluated on elements of cohomological degree 1, hence the given formula indeed suffices to

describe it. To arrive at the formula one simply computes the commutator of ια + ∇X and

ιβ +∇Y , viewed as derivations of SymA.

Now we compute the de Rham differential. First, we remark that

δ :=∑

ιei ⊗ ei

is the standard Koszul differential for the algebra SymA, expressed in local basis ei of A.

Moreover, the curvature R∇ ∈ ∧2T∗M⊗A⊗A∨ of the connection naturally induces a morphism

R∇ : ∧kA −→ Sym1 A⊗∧k−1A⊗∧2T∗M.

To prove the formulas for the de Rham differential, one uses that it is defined by

dA∨(g)(V1) = V1(g)

dA∨(γ)(V1, V2) = V1(γ(V2)) − V2(γ(V1)) + γ([V1, V2]SN)

where g ∈ SymA, Vi ∈ L and γ ∈ Ω1L(M). The fact that the de Rham differential squares to

0 is equivalent to the Bianchi identity. Finally, the Lie algebra module structure on ΩL(M) is

determined in low degrees because it is compatible with the wedge product on forms and the

identity [Lγ1, Lγ2

] = L[γ1,γ2] for γi ∈ TLpoly(M). The given formulas are all direct consequences

of the following expression:

LV(γ) = (ιγdA∨ + dA∨ιγ)V,

where V ∈ L and γ ∈ Ω1L(M).

3.1.2 The Poisson structure on A∨

From now on we assume that π : A → M is a smooth Lie algebroid. It is well-known that

the Lie algebroid structure on A defines a canonical Poisson structure on the total space of the

dual vector bundle A∨ by the following formula:

π∗f, π∗g := 0, s, π∗f := π∗(ρ(s)(f)), s, t := [s, t], f ∈ C∞(M), s, t ∈ Γ(A). (3.1.6)




This defines the Poisson bracket on functions which are constant or linear along the fibers.

These functions generate the functions which are polynomial along the fibers, which are in turn

a dense subalgebra of the smooth functions on A∨, so the Poisson bracket can be extended to

C∞(A∨) using the Leibniz rule.

The properties of the Lie bracket and the anchor map imply that Equation (3.1.6) defines

a Poisson structure, i.e. that it satisfies the Leibniz and Jacobi identities.

Recall that a Poisson structure on A∨ is equivalent to an element θ ∈ T2(A∨) such that

[θ, θ]SN = 0. The relation to the Poisson structure is given by

h, k = θ(dh, dk), h, k ∈ C∞(A∨). (3.1.7)

Remark 3.1.3. We differ from the usual sign convention for the Poisson tensor defined on A∨,

since it seems more natural for most of our constructions.

We view the Poisson tensor as element in TL,1poly(A

∨), i.e. as a multiderivation of SymA. It

is important to note that we used the shifted degree on the space of polyvectorfields; i.e. its

cohomological degree as in (3.1.4) is equal to 2. The shifted degree, which gives the bracket

[ , ]SN degree 0, is conventional in the theory around formality that we will use later.

Given a connection ∇ on A and s, t ∈ Γ(A), the formula (s, t) → ∇ρ(s)t defines an A-

connection ∇ρ( ) on A. The A-torsion TA∇ of ∇ρ( ) is defined by the following equation:

TA∇ : s∧ t → [s, t] −∇ρ(s)t+∇ρ(t)s.

We view it as an element in Sym1 A ⊗ ∧2A∨. Also, we view the anchor ρ as an element of

∧1A∨ ⊗∧1TM.

Remark 3.1.4. In [NeWa] the authors note that one can shift A-connections on A by half of

the torsion to obtain an A-torsion free A-connection. However, the torsion free A-connection

associated to an A-connection of the form ∇ρ( ) does not in general have the form ∇ρ( ) for a

TM-connection ∇ on A.

Lemma 3.1.5. The Poisson tensor defined by the Equations in (3.1.6) is, under the isomor-

phism

TL,1poly(A

∨)∇∼= SymA⊗ (∧2A∨ ⊕ (∧1A∨ ⊗∧1TM)⊕∧2TM),

given by θ∇ := TA∇ + ρ.

Proof. We use the identification of (3.1.7). Let f, g ∈ C∞(M) and s, t ∈ Sym1 A. It follows

easily that

θ∇(d∇f, d∇g) = θ∇(dMf, dMg) = 0.

Moreover we, have

θ∇(d∇s, d∇f) =θ∇(δ(s) +∇s, dMf)

=ρ(s)(dMf)

θ∇(d∇s, d∇t) =θ∇(δ(s), δ(t)) + θ∇(∇s, δ(t))

+θ∇(δ(s),∇t) + θ∇(∇s,∇t)

=[s, t] −∇ρ(s)t+∇ρ(t)s−∇ρ(t)s+∇ρ(s)t

=[s, t].

Comparing this to the Poisson structure defined by the Equations in (3.1.6), the lemma is

proved.




The Leibniz and Jacobi identities for the Poisson bracket, which are concisely captured by

the Maurer–Cartan equation

[θ, θ] = 0,

imply, together with the Jacobi identity for the Schouten–Nijenhuis bracket on polyvectorfields,

that the map

[θ, ] : TL,kpoly(A

∨) −→ TL,k+1poly (A∨), γ → [θ, γ]

squares to 0, and this complex defines the so-called Poisson cohomology. This is explained in

more detail in subsection 3.1.4.

3.1.3 Representations up to homotopy

Let us recall that representations of Lie algebroids A → M are given by flat Lie algebroid

connections on vector bundles E → M. The major drawback of this definition of a representation

is its rigidity, so in general, the category of representations of a Lie algebroid is quite small,

and in particular may not contain anything that can be called the adjoint representation. This

can already be seen in the case of the tangent bundle; any vector bundle which has non zero

characteristic classes does not admit flat connections, so cannot be a TM-representation. As a

remedy, the more flexible notion of a representation up to homotopy was introduced in [AC],

which was based on earlier ideas of, for example, [ELW, CM, Qu]. In this section, we shortly

describe how a representation up to homotopy is defined, for a more thorough treatment we

refer to [AC]. Informally speaking, the vector bundle E is replaced by a complex of vector

bundles . . .∂→ Ei ∂→ Ei+1 ∂→ . . . , and the flat A-connection on E is replaced by not necessarily

flat connections on each bundle Ei which are compatible with the maps ∂, i.e. ∇∂ − ∂∇ = 0.

The data is completed by a collection of tensors satisfying relations which imply that, together

with ∂ and ∇, they form a derivation D of degree 1 which squares to zero on the total complex

ΩA(⊕

i Ei), which is of course analogous to the fact that flat A-connections on a vector bundle

E induce differentials on ΩA(E). Since we deal with graded vector bundles, plenty of signs are

introduced, but in fact they all boil down to the usual Koszul sign rule, which says that for any

transposition of elements x, y of degree |x|, |y|, a sign (−1)|x||y| is introduced.

Definition 3.1.6. Given a smooth Lie algebroid A → M, a representation up to homotopy of

A is given by a pair (E,D), where E =⊕

i∈Z Ei is a graded vector bundle, and a map

D : ΩA(E) −→ ΩA(E)

which increases the total degree by 1, squares to 0 and is a graded derivation, i.e.

D(ω∧ η) = dA(ω)∧ η+ (−1)kωD(η), for ω ∈ ΩkA, η ∈ ΩA(E).

The cohomology of the resulting complex is denoted by H•(A;E).

With respect to the degree in E, we can decompose D as a sum

D = ∂+∇+∑i2

ωi,

where:

1. The map ∂ is a degree 1 operator on E turning E into a complex.

2. The map ∇ is an A-connection on E such that [∇, ∂] = 0.

3. The maps ωi are given by tensors ωi ∈ ΩiA(End

1−i(E)), with the usual correspondence

between such tensors and ΩA-module morphisms ΩkA(E

j) → Ωk+iA (Ej+1−i).




These maps satisfy, in addition to ∂2 = 0 and [∇, ∂] = 0, the relations ∂ ω2 + R∇ = 0 and

∂ ωi +∇ωi−1 +ω2 ωi−2 + . . .+ωi−2 ω2 = 0

Example 14 (The adjoint representation, [AC]). The graded vector bundle of the adjoint

representation up to homotopy is given by A⊕ TM, where A has degree 0 and TM has degree

1. Given a TM-connection on the vector bundle A → M the associated basic A-connection on

A⊕ TM is defined by the formulas

∇bass t :=∇ρ(t)s+ [s, t]

∇bass X :=ρ(∇Xs) + [ρ(s), X]

where X ∈ Γ(TM) and s, t ∈ Γ(A). The basic curvature of the connection is defined by

Rbas∇ (s, t)(X) = ∇X([s, t]) − [∇Xs, t] − [s,∇Xt] −∇∇bas

t X(s) +∇∇bass X(t).

A straightforward check shows that this is a tensor Rbas∇ ∈ Ω2

A(Hom(TM,A)). In [AC] the

following proposition was proved:

Proposition 3.1.7. The basic curvature Rbas∇ of a TM-connection ∇ on A satisfies the following

identities:

1. For the curvature R∇bas of the A-connection ∇bas on A one has Rbas∇ ρ = R∇bas , and for

the curvature R∇bas of the A-connection ∇bas on TM one has R∇bas = −ρ Rbas∇ .

2. The basic curvature is closed with respect to the differential d∇bas on ΩA(Hom(TM,A)),

i.e. d∇basRbas∇ = 0.

The differential of the adjoint representation is now given by

D∇ = ρ+∇bas + Rbas∇ .

The previous proposition together with the property ρ ∇bas = ∇bas ρ are equivalent to the

fact that D2∇ = 0. The complex thus obtained will be denoted by (Ω•

A(Ad(A)), D∇).

Example 15 (The double representation [AC]). Let E → M be a vector bundle. Any A

connection on E defines a representation of A on Eid−→ E, where the copies of E have degree 0

and 1 by setting

D = id+∇+ R∇.

This representation, called the double of a vector bundle, is denoted by DE.

As mentioned in the introduction, one can define tensor products of representations. More

generally, one can define direct sums, wedge products, duals, symmetric powers and, given two

representations up to homotopy E and F, the representation Hom(E, F). With these operations

one can construct a wealth of representations up to homotopy, which is done in [AC], but

here we only discuss a few examples which are related to the polynomial polyvector fields and

differential forms. A particular operation, which we will need later, is the following.

Example 16 (Conjugation [AC]). Given a representation up to homotopy E with operator

D, on can form a representation with the same underlying vector bundle E, and the structure

operator given by

D = −∂+∇−ω2 +ω3 − . . . .

This representation is isomorphic to the original one under the map Φ given by (−1)nId on En.




Example 17 (Symmetric powers). The k-th symmetric power of the adjoint representation of

the Lie algebroid A → M from example 14 gives rise to the following complex of vector bundles:

0 −→ Symk A −→ Symk−1 A⊗ TM −→ . . . −→ Sym1 A⊗∧k−1TM −→ ∧kTM −→ 0,

where the degree of Symk−i A ⊗ ∧iTM is i. The differential D of the representation acts

on Symi A by the extensions of ρ and ∇bas by derivations, on ∧jTM by the extensions

of ∇bas and Rbas∇ by derivations, and on ∧lA∨ by dA. We denote the representation by

ΩA(Symk(adA)), D∇). The direct sum of representations ΩA(

⊕k Sym

k adA), D∇) admits

a product which is a combination of the symmetric product on SymA, the wedge product

on ∧TM and the wedge product on ∧A∨. It is clear from the form of the differential that

(ΩA(⊕

k Symk adA),∧, D∇) =: (ΩA(SymadA),∧, D∇) is a differential graded algebra.

Example 18 (Symmetric powers of the double). The k-th symmetric power of the double

vector bundle representation Eid−→ E of A from example 15 give rise to the complex

0 −→ Symk E −→ Symk−1 E⊗ E −→ . . . −→ Sym1 E⊗∧k−1E −→ ∧kE −→ 0

of vector bundles, where the differential of the complex is given by the extension of the identity

map on E by derivations to Symi E, which turns out to the be standard Koszul differential for

the bundle SymE⊗∧•E. The part of D which preserves the degree of the graded vector bundle

is simply given by the extension of the connection ∇ to Symi E and ∧jE, whereas the part of D

in Ω2A(End

−1(Symk(Eid−→ E))) is given by the extension of the curvature R∇ ∈ Ω2

A(End(E))

to a map R∇ : Symi E ⊗ ∧jE −→ Symi+1 E ⊗ ∧j−1E ⊗ ∧2A∨, which defines an element in

Ω2A(End

−1(Sym(Eid−→ E))). The direct sum of representations Sym(E

id→ E) :=⊕

k Symk(E

id→E) again is a differential graded algebra.

3.1.4 Poisson (co)homology and the modular class

In this section we recall some facts about Poisson manifolds and Lie algebroids, and in particular

the definition of the modular class. The material is standard and can for example be found in

[Br], [ELW] or [CdSW]. Let P be a Poisson manifold of dimension n, i.e., a manifold together

with a Poisson tensor π ∈ Λ2TP such that [π, π]SN = 0. The Poisson cohomology complex of P

is given by the polyvector fields on P with differential ∂π := [π, ], and the resulting cohomology

groups are denoted by H•π(P). The differential forms Ω(P) over P form a module over the

polyvector fields, and this enables one to define Lπ : Ω•(P) → Ω•−1(P), which is the Poisson

homology differential. The resulting homology groups are denoted by Hπ• (P). For simplicity we

assume that P is orientable i.e., there exists a non-vanishing global volume form µ ∈ Ωn(P) on

P. The volume form induces an isomorphism

µ : Λ•TP∼=−→ Ωn−•(P), γ → ιγµ (3.1.8)

which can be used to define the modular vector field of P, by comparing the Poisson cohomology

differential to the Poisson homology differential transferred to the polyvector fields, denoted by

µ(Lπ) := µ−1 Lπ µ.

Definition 3.1.8. Given a Poisson manifold P with a non-vanishing volume form µ ∈ Ωn(P),

the modular vector field νµ is defined by

(µ(Lπ) − ∂π)γ = νµ ∧ γ.

Proposition 3.1.9. The modular vector field νµ of a Poisson manifold associated to a volume

form µ is well-defined. Moreover, it can equivalently be characterized by:




i) Given a function f ∈ C∞(P) and its Hamiltonian vector field Xf, one has

LXfµ = νµ(f)µ.

ii) The following relation between the Poisson tensor, the volume form, and the modular

vector:

Lπµ = ινµµ.

Given two volume forms µ1 and µ2 for which µ1 = fµ2 holds for f ∈ C∞(P), one has

νµ2= νµ1

− Xlog |f|,

where Xlog |f| is the Hamiltonian vector field of log |f|. The modular vector field satisfies ∂π(νµ) =

0, hence it defines a class [ν] ∈ H1π(P) in the Poisson cohomology, called the modular class.

Proof. Although this lemma is often stated in the literature, explicit proofs are not so abundant,

therefore we decided to give some details. First we prove that the first characterization defines

a vector field. To do this we show that it is a derivation:

LXfgµ =LfXg+gXfµ = (fLXg + gLXf + df∧ LXg + dg∧ LXf)µ

=(fLXg + gLXf + LXg(df) + LXf(dg))µ

=(fLXg + gLXf)µ

where we used the antisymmetric property of the Poisson tensor in the last line. The following

equalities show that the first and second characterization are equivalent:

LXfµ =− dιιdfπµ = −dιπ(df∧ µ) + d(df∧ ιπ)µ = df∧ Lπµ

ν(f)µ =ιν(df)µ = df∧ ινµ.

The next computation shows that assertion ii) from the proposition and the definition of the

modular vector field are equivalent:

µ(∂πγ+ νµ ∧ γ) =ι[π,γ]µ+ ιν∧γν

=Lπιγµ− ιγLπµ+ ιγινµ

=Lπιγµ− ιγLπµ+ ιγLπµ

=Lπ(µ(γ)).

Now let µ1 = fµ2 where µi ∈ Ωn(P) are non-vanishing. Then we have

Lπ(fµ1) = dιπ(fµ1)

= df∧ ιπµ1 + fLπµ1

= ιπ(df∧ µ1) + ιιdfπµ1 + fLπµ1

= −ιXfµ1 + fLπµ1.

Hence it follows that νµ2= νµ1

− X| log f|. For the last part, we compute

ι[π,ν]µ = (Lπιν + ινLπ)µ = (L2π + ι2ν)µ = 0.

In the same way as we did with the Poisson homology differential, we can transfer the de

Rham differential to the polyvectorfields, i.e. µ(d) = µ−1 d µ. The second characterization

from the last proposition is then also equivalent to

µ(d)(π) = νµ.




The modular class of a Poisson manifold is in fact part of a more general story for Lie algebroids,

which we will describe now.

As mentioned in the discussion about representations up to homotopy, Lie algebroids admit

few representations, i.e. flat A-connections on vector bundles. The following construction, in

which a canonical representation on a line bundle is constructed, provides an exception to this

rule. Representations on line bundles give rise to characteristic cohomology classes in H1(A).

In the special case that we consider, these classes are particular examples of the secondary

characteristic classes considered in, for example, [CF2]. Let A → M be a smooth Lie algebroid

of rank r over a manifold of dimension n. We define the line bundle

QA := ∧rA⊗∧nT∗M.

For simplicity, let us assume that it admits a non-vanishing global section T ⊗ µ.

Lemma 3.1.10 ([ELW]). The equation

∇QAs (T ⊗ µ) := [s, T ]⊗ µ+ T ⊗ Lρ(s)µ

for s ∈ Γ(A) defines a representation of A on QA.

Given a (trivial) line bundle L over M together with a flat A-connection D on L, one can

define the characteristic class [µL] ∈ H1(A) of this representation by the equation

Dsl = µlL(s)l

where s ∈ Γ(A), l ∈ Γ(L) is non-vanishing and µlL ∈ Γ(A∨). The fact that µl

L ∈ Γ(A∨) is

straightforward, whereas dAµlL = 0 follows directly from the fact that D is flat. Given two

non-vanishing sections l1, l2 ∈ Γ(L) which are related by fl1 = l2, an easy computation shows

that µl2L (a) = µl1

L (a) + d(log |f|)(a), so up to exact forms the definition is independent of the

choice of a section, hence [µL] ∈ H1(A) is well-defined. Choosing L = QA, this construction

gives the modular class of the Lie algebroid A. The following proposition gives an interpretation

of the characteristic class of a Lie algebroid which is similar to Definition 3.1.8.

Proposition 3.1.11. Given a representation of A on a trivial line bundle L, the element

µlL ∈ Γ(A∨) associated to the non-vanishing section l ∈ Γ(L) satisfies

dA + µlL∧ = Dl

where Dl is the differential on ΩA obtained by transferring dD on ΩA(L) to ΩA by the iso-

morphism

ΩA

∼=−→ ΩA(L), α → α⊗ l.

Proof. Writing dD = D : Γ(L) −→ Ω1A(L), we see that D(l) =: µL⊗l ∈ A∨⊗L. Then the claim

follows directly from the general requirement for the de Rham differentials with coefficients that

dD(α⊗ l) = dA(α)⊗ l+ α∧ dD(l).

Let P be a Poisson manifold and A = T∗P the Lie algebroid associated to P. There are two

definitions of the modular class; one for the Poisson manifold P and one for the Lie algebroid

T∗P. These are related as follows.

Proposition 3.1.12 ([ELW]). Given a Poisson manifold P, the modular class θP of the Poisson

manifold and the modular class θT∗P of the Lie algebroid T∗P satisfy

θP =1

2θT∗P.

Moreover, the bundle ∧topT∗P admits a representation which characteristic class is equal to the

modular class of P.




Let us remark that QT∗P∼= (∧topT∗P)2. The second part of the proposition is a consequence

of the fact that, given a representation on the square of a line bundle L one can define a

representation on L itself. Given µ ∈ Γ(∧topT∗P) and α ∈ Γ(T∗P), the representation of T∗P on

∧topT∗P constructed in this way is given by:

Dαµ =[α, µ]T∗P − π(dα)µ (3.1.9)

=α∧ Lπµ. (3.1.10)

The de Rham complex associated with this representation is given by

(Ω•T∗P(∧

nT∗P), dD) = (Γ(∧•TP ⊗∧nT∗P), dD)

where dD satisfies, given γ ∈ Γ(∧kTP),

D(γ⊗ µ) =dT∗Pγ⊗ µ+ (−1)kγ∧Dµ)

=[π, γ]SN ⊗ µ+ (−1)kγ∧Dµ)

The contraction γ⊗ µ → ιγ(µ) defines an isomorphism

τ : Ω•T∗P(∧

nT∗P)∼=−→ Ωn−•(P)

and this isomorphism leads to the following theorem:

Theorem 3.1.13 ([ELW], Thm. 4.5). For γ⊗ µ ∈ Ω•T∗P(∧

nT∗P) one has

τ(dD(γ⊗ µ)) = (−1)•+1Lπ(τ(γ⊗ µ)),

hence H•Lie(T

∗P,∧nT∗P) ∼= Hπn−•(P).

3.1.5 Poisson (co)homology vs. representations up to homotopy

In this subsection we finally relate the polynomial Poisson complexes of the dual of a Lie

algebroid A → M to representations up to homotopy. Recall the Lie algebroid L = DerR(SymA)

over (M, SymA), its polyvector fields and differential forms, and the Poisson (co)homology

differentials.

Theorem 3.1.14. Given a Lie algebroid A → M and a connection ∇ on A, we have the

following isomorphism:

Hθ• (T

Lpoly(A

∨)) ∼= H•(A, Sym(adA)).

Theorem 3.1.15. Let A → M be an orientable smooth Lie algebroid of rank r over an ori-

entable manifold M of dimension n, then we have the following isomorphism:

Hn+r−•θ

(Ω•

L(A∨)

)∼= H• (A,Sym(adA)⊗QA) .

Before giving the proofs of these theorems, we prove a simple lemma that, given a vector

bundle E → M, relates the polynomial differential forms on E∗ with the de Rham differential

to the representation up to homotopy from example 18 in the case A = TM.

Lemma 3.1.16. Given a vector bundle E → M and an affine connection ∇ on E, the de

Rham differential dSymE of the polynomial differential forms ΩSymE is equal to D∇ of the

representation Sym(Eid−→ E) of example 18 under the isomorphism induced by the connection

∇ of (3.1.5).

Proof. Recall from lemma 3.1.2 that the de Rham differential d∇ on SymA ⊗ ∧A ⊗ ∧T∗M,

which has polynomial degree 0 and cohomological degree 1, is given by:




i) The sum of the Koszul differential and the connection on the Sym(E) part.

ii) The de Rham differential on the ∧T∗M part.

iii) The sum of the connection and the curvature R∇, considered as a map R∇ : ∧kA →Sym1 A⊗∧k−1A∧2 T∗M.

This exactly agrees with the discussion of the differential D∇ of the symmetric power of the

double vector bundle representation Eid−→ E in the last example.

The following lemma not only proves theorem 3.1.14, but is in fact a stronger statement since

it gives an isomorphism between the two complexes on the level of cochains. This isomorphism

depends on the choice of a connection on A.

Lemma 3.1.17. Given a Lie algebroid A → M and a connection ∇ on A, the Poisson cohomol-

ogy complex (TLpoly(A

∨), [θ, ]) is isomorphic to the symmetric power of the adjoint representation

(ΩA(Sym(adA)), D∇).

Proof. We will prove that, under the identification

TLpoly(A

∨)∇∼= SymA⊗∧A∨ ⊗∧TM

the Poisson differential [θ, ] translates to D∇, the differential of the conjugate representation

of Sym(adA). Since representations are isomorphic to their conjugates this proves the lemma.

Note that this lemma is a natural extension of lemma 3.1.2. The differential [θ, ]SN on TLpoly(A

∨)

has polynomial degree −1 and cohomological degree 1, hence, under the isomorphism induced

by ∇ it gives a map

[θ∇, ]SN :⊕

s+t=jr−s=k

Symr A⊗∧sA∨ ⊗∧tTM −→⊕

s+t=j+1r−s=k−1

Symr A⊗∧sA∨ ⊗∧tTM

where j and j + 1 indicate the cohomological degree and k and k − 1 indicate the polynomial

degree. Recall from lemma 3.1.5 that the Poisson tensor θ ∈ TL,2(A∨) corresponds to TA∇ + ρ

with TA∇ ∈ Sym1 A⊗∧2A∨ and ρ ∈ ∧1A∨ ⊗∧1TM under the isomorphism induced by ∇.

We have to prove that D∇ = [θ∇, ]SN. Let us give two simple but useful formulas from the

Poisson cohomology complex. Let g ∈ SymA, γ ∈ TL,1(A∨) and ωi ∈ Ω1L(A

∨), then

[θ, g](ω1) = − θ(dg,ω1)

[θ, γ](ω1,ω2) = − Lγθ(ω1,ω2)

= − γ(θ(ω1,ω2)) + θ(Lγω1,ω2) + θ(ω1, Lγω2)

(3.1.11)

To compute [θ∇, ]SN, we evaluate it on one forms using the formula above and the expressions

for the de Rham differential and the Lie bracket from 3.1.2. Let s, t ∈ Sym1 A, α ∈ ∧1A∨ and

X, Y ∈ ∧1TM.

• We start with computing the action of [θ∇, ] on SymA. This is clearly determined by

the action on Sym1 A, thus we have to compute [ρ, s] ∈ Sym1 A ⊗ ∧1A∨ ⊕ ∧1TM and

[TA∇ , s] ∈ Sym1 A⊗∧1A∨:

[ρ, s](ξ) = − ρ(d∇s, ξ) = −ρ(δs+∇s, ξ) = −ξ(ρ(s))

[ρ, s](t) = − ρ(d∇s, t) = −ρ(δs+∇s, t) = ∇ρ(t)s

[TA∇ , s](t) = − TA

∇ (d∇s, t) = −TA∇ (δs, t) = −[s, t]A +∇ρ(s)t−∇ρ(t)s.

These equalities show that [θ∇, s] = −ρ(s) +∇bass.




• Now we compute the action of [θ∇, ] on polynomial vector fields. Let α ∈ ∧1A∨.

From the expression for the Schouten–Nijenhuis bracket it follows easily that [θ∇, α] ∈Sym1 A⊗∧2A∨. The equalities

[ρ, α](s, t) = −ια(ρ(s, t)) + ρ(Lα(s), t) + ρ(s, Lα(t))

= ρ(∇α(s), t) + ρ(s,∇α(t))

= −ρ(t)(α(s)) + α(∇ρ(t)s) + ρ(s)(α(t)) − α(∇ρ(s)t)

[TA∇ , α](s, t) = −ια([s, t]∇) + [Lαs, t]∇ + [s, Lαt]∇

= −α([s, t]) + α(∇ρ(s)t) − α(∇ρ(t)(s))

(3.1.12)

imply that [θ∇, α] = dAα, where dA is the de Rham differential of ΩA.

• Finally, let us compute [θ∇, X]. This time, the expression for the Schouten–Nijenhuis

bracket implies that [θ∇, X] ∈ (∧1A∨ ⊗ ∧1TM) ⊕ (Sym1 A ⊗ ∧2A∨). We assume for

simplicity that ξi = d∇fi = dMfi for fi ∈ C∞(M).

[ρ, X](s, d∇f) = − X(ρ(s, d∇f) + ρ(LXs, d∇f) + ρ(s, LX(d∇f)

= − X(ρ(s)(f)) + ρ(∇Xs+ R(X)(s), d∇f) + ρ(s, d∇(X(f))

= [ρ(s), X](f) + ρ(∇Xs)(f)

[ρ, X](s, t) = ρ(−R∇(X)(s), t) + ρ(s,−R∇(X)(t)))

=R∇(X, ρ(t))(s) − R∇(X, ρ(s))(t)

=([∇X,∇ρ(t)] −∇[X,ρ(t)]

)s−

([∇X,∇ρ(s)] −∇[X,ρ(s)]

)t

[TA∇ , X](s, t) = −∇X(T

A∇ (s, t)) + TA

∇ (LXs, t) + TA∇ (s, LXt)

= −∇X([s, t]) +∇X∇ρ(s)t−∇X∇ρ(t)s)

+ [∇Xs, t] −∇ρ(∇Xs)t+∇ρ(t)∇Xs

+ [s,∇Xt] −∇ρ(s)∇Xt+∇ρ(∇Xt)s

=−∇X([s, t]) + [∇Xs, t] + [s,∇Xt]

+ [∇X,∇ρ(s)]t− [∇X,∇ρ(t)]s

+ ∇ρ(∇Xt)s−∇ρ(∇Xs)t.

The first equality implies that [θ∇, X]|∧1A∨⊗∧1TM = ∇basX. Moreover, a careful analysis

of the signs in the second and third equalities reveal that

[θ∇, X](s, t) = −Rbas∇ (s, t)(X).

This finalizes, using the expression for D∇ from example 17, the proof that D∇ coincides with

[θ∇, ].

The last part of this section is devoted to a proof of theorem 3.1.15. Again, this follows

from a stronger statement on the level of chains:

Lemma 3.1.18. Given a smooth, orientable Lie algebroid A of rank r over an orientable

manifold M of dimension n, we have the following isomorphism:

(Ωn+r−•

L (A∨), Lθ)∼=(Ω•

A(SymadA⊗QA), D∇

)

where D∇ is the differential of the tensor product of the representations SymadA and QA.

Loosely speaking, the way to prove this lemma is to:

• use the identification

ΩL(A∨)

∇∼= SymA⊗∧A⊗∧T∗M,




• and relate the representation of A on QA and the representation of Ω1L(A

∨) on ΩtopL (A∨)

to each other, which also implies that the modular class of the Lie algebroid A lifts in a

natural way to the modular class of the Poisson manifold A∨.

The first part is already extensively described in lemma 3.1.2 and 3.1.17, so we focus on the

second part. Recall that the connection on A induces

L∗ = Ω1L(A

∨)∇∼= SymA⊗ (A⊕ T∗M). (3.1.13)

Consider the following maps from A into this bundle:

d∇ : Γ(A) −→ SymA⊗ (∧1A⊕∧1T∗M) s → d∇(s) = δs+∇s

δ : Γ(A) −→ SymA⊗ (∧1A⊕∧1T∗M) s → δs.

The first map is given by s → dA∨(s) under the identification from (3.1.13). Note that δs is the

Koszul differential of SymA applied to s ∈ Sym1 A, and one has δs = s ∈ ∧1A. The following

lemma will be used throughout this section.

Lemma 3.1.19. The natural inclusion (Ω•A, dA) −→

(TLpoly(A

∨), [θ, ])defined by α → ια for

α ∈ Γ(A∨) is a morphism of differential graded algebras.

Proof. This is a direct consequence of formula 3.1.12.

As we will see in the following two lemmas, the maps d∇ and ι both respect only part

of the structures, i.e. they are not morphisms of Lie algebroids. In the first lemma we will

view Γ(A) as a Lie algebra together with a Lie algebra action on C∞(M), and similarly we will

view L∗ = Ω1L(A

∨) as a Lie algebra (with the Lie bracket induced from the Poisson structure

θ ∈ TL,2poly(A

∨)) with a Lie algebra action on SymA. We will write dA∨ = d for aesthetic

reasons.

Lemma 3.1.20. The following diagrams define morphisms of Lie algebras and their modules:

Γ(A) C∞(M) Γ(A) Γ(A)

L∗ SymA L∗ SymA

ρA

d ι

ad

d ι

ρL∗ ρL∗

(3.1.14)

Proof. We will not include ι in the notation since it is clear from the context where it should

be written. The fact that d is a morphism follows from the definition of the Poisson structure

on A∨:

[ds, dt]L∗ = ds, t = d([s, t]).

The fact that the first diagram is commutative also follows from the definition:

ds(f) = θ(ds, df) = s, f = ρ(s)(f),

and the equation

ρL∗(ds)(t) = θ(ds, dt) = [s, t]

proves the fact that the second diagram is commutative.

Lemma 3.1.21. The inclusion δ is C∞(M)-linear and satisfies

[δs, δt]L∗ = δ([s, t]A) + Rbas∇ (s, t)

ρL∗(δs)(f) = ρ(s)(f).




Recall that, given two one forms α1, α2 ∈ Ω1L(A

∨), the Lie algebroid structure on Ω1L(A

∨)

is defined by

[α1, α2]L∗ = −dθ(α1, α2) + Lθ(α1, )α2 − Lθ(α2, )α1

= dθ(α1, α2) + ιθ(α1, )dα2 − ιθ(α2, )dα1.

Given s, t ∈ Γ(A), we will compute [δ(s), δ(t)]L∗ −δ([s, t]A) using the above formula. We know

from the formulas for θ, the Lie derivatives and d∇ that

[δ(s), δ(t)]L∗ − δ([s, t]A) ∈ ∧1A⊕ (Sym1 A⊗∧1T∗M),

so we can evaluate in β ∈ Γ(A∨) or X ∈ Γ(TM). First we compute

dθ(δ(s), δ(t))(β) = β([s, t] −∇ρ(s)t+∇ρ(t)s)

ιθ(s, )dt(β) = dt(ρ(s), β) = β(∇ρ(s)t)

−ιθ(t, )ds(β) = −ds(ρ(t), β) = −β(∇ρ(t)s)

from which it follows that ([δ(s), δ(t)]L∗ − δ([s, t]A)) |∧1A = 0. Then we evaluate in X:

dθ(δ(s), δ(t))(X) = ∇X([s, t] −∇ρ(s)t+∇ρ(t)s)

ιθ(s, )dt(X) = ιρ(s)dt(X) + ι[s, ]∇dt(X)

= R(X, ρ(s))(t) − ([s,∇Xt] +∇ρ(s)∇Xt+∇ρ(∇Xt)s)

−ιθ(t, )ds(X) = −R(X, ρ(t))(s) + ([t,∇Xs] +∇ρ(t)∇Xs+∇ρ(∇Xs)t)

from which it follows that ([δ(s), δ(t)]L∗ − δ([s, t]A)) |Sym1 A⊗∧1T∗M = Rbas∇ (s, t).

Remark 3.1.22. The previous lemma is related to [AC][Section 3.2]. The authors consider

the short exact sequence for the jet bundle of A

0 −→ Hom(TM,A) −→ J1(A) −→ A −→ 0.

This is the degree 1 part of the short exact sequence for the polynomial differential forms

that is induced by the connection. The splitting j on the level of sections which is defined by

s → j(s) is not C∞(M)-linear, and corresponds in our story to the map induced by the de Rham

differential. In [AC], the Lie algebra structure on J1(A) is defined by requiring j to be a Lie

algebra morphism, which is similar to our definition of the Lie algebroid structure on Ω1L(A

∨).

The splitting j∇ = j + ∇ of the short exact sequence for the jet bundle is C∞(M)-linear and

corresponds to our map δ. This map does not respect the Lie algebra structures on Γ(A) and

Γ(J(A)), but satisfies the same equation involving the basic connection.

The maps d and δ can both be extended to maps

d : QA −→ ΩtopL (A∨)

δ : QA −→ ΩtopL (A∨)

defined by

t1 ∧ · · ·∧ tr ⊗ µ1 ∧ · · ·∧ µn −→ d(t1)∧ · · ·∧ d(tr)⊗ µ1 ∧ · · ·∧ µn

t1 ∧ · · ·∧ tr ⊗ µ1 ∧ · · ·∧ µn −→ δ(t1)∧ · · ·∧ δ(tr)⊗ µ1 ∧ · · ·∧ µn

where ti ∈ Γ(A), µi ∈ T∗M. These two maps are equal because the terms involving ∇(ti) ∈Sym1 A ⊗ ∧1T∗M cancel for degree reasons. We will denote this map by d. Recall that QA

is an A-representation with the flat connection denoted by ∇QA = D, and Ωn+rL (A∨) is a

representation of the Lie algebroid Ω1L(A

∨) with connection D.




Lemma 3.1.23. The following diagram defines a morphism of Lie algebras and their modules:

Γ(A) Ω1L(A

∨)

QA Ωn+rL (A∨)

D

d

D

d

Proof. Recall that the action of A on QA is given by

Ds(T ⊗ µ) = [s, T ]SN ⊗ µ+ T ⊗ Lρ(s)µ

where s ∈ Γ(A), T ∈ ∧rA, µ ∈ ∧nT∗M. To compute the action of Ω1L(A

∨) on Ωn+rL (A∨) we

work locally, and write T = t1 ∧ · · · ∧ tr and µ = f0df1 ∧ · · · ∧ dfn for fi ∈ C∞(M). Given

g ∈ SymA, the action of Ω1L(A

∨) on a section Ω ∈ Ωn+rL (A∨) has the form

DdgΩ = [dg,Ω]L∗ = Lθ(dg, )Ω

which follows from the fact that the Schouten–Nijenhuis bracket on ΩL(A∨) has the property

that [dg, dh]L∗ = d(θ(dg, dh)) = Lθ(dg, )dh where g, h ∈ SymA. Let s, t ∈ Γ(A), then

Lθ(ds, )dt = d(θ(ds, dt)) = d([s, t])

from which we can conclude that Lθ(ds, )(dt1 ∧ · · · ∧ dtr) = d ([s, t1 ∧ · · ·∧ tr]A). Moreover,

we have

Lθ(ds, )df = d(θ(ds, df)) = d(ρ(s)(f)) = Lρ(s)df

from which we can conclude that Lθ(ds, )(f0df1∧ · · ·∧dfn) = Lρ(s)(f0df1∧ · · ·∧dfn). Finally

we have

Lθ(ds, )(d(T ⊗ µ)) = d([s, T ]SN ⊗ µ+ t⊗ Lρ(s)µ

)= d(Ds(t⊗ µ)),

which proves the lemma.

Corollary 3.1.24. Given a non vanishing section Ω ∈ QA, the modular form νΩ ∈ Ω1A

associated to the Lie algebroid A satisfies

dνΩ = νdΩ,

where νdΩ is the modular vector field of the Poisson structure on A∨ associated to the volume

form dΩ.

Proof. First we remark that, given α ∈ Γ(A∨) and s ∈ Γ(A), one has 〈ds, dα〉 = α(s) ∈ Sym0 A

where 〈·, ·〉 is the natural pairing between L∗ and L.

Using the last lemma and the definitions for the modular vector fields of Poisson structures

and Lie algebroids, we compute:

〈ds, dνΩ〉dΩ =d(νΩ(s)Ω) = d(Ds(Ω)) = Dds(dΩ) = Lθ(ds, )dΩ = 〈ds, νdΩ〉dΩ

which proves the lemma.

Proof of lemma 3.1.18. The differential of the representation Sym(adA) ⊗ QA is defined by

D∇ ⊗ 1+ 1⊗D, where D∇ is the differential of Sym(adA) and D defines the representation of

A on QA. From lemma 3.1.17 we know that under the isomorphism

Ω•A(Sym(adA))

∇∼= TL

poly(A∨)



3.2. APPLICATION OF FORMALITY 63

induced by the connection the differential D∇ corresponds to [θ, ] up to signs, and from lemma

3.1.23 we know that the action of A on QA and the action of L∗ on Ωn+rL (A∨) are intertwined

under the maps d : A −→ L∗ and d : QA −→ Ωn+rL (A∨). It follows that

(Ω•

A(Sym(adA)⊗QA), D∇

) ∇∼=(TL,•pol (A

∨)⊗Ωn+rpol (A∨), [θ, ]⊗ 1+ 1⊗Dθ

)

is an isomorphism of differential graded algebras, where D∇ is the differential for the represen-

tation Sym(adA) ⊗ QA. Finally, the composition of this isomorphism with the isomorphism

from 3.1.13 leads to

(Ω•

A(Sym(adA)⊗QA), D∇

) ∇∼=(Ωn+r−•

pol (A∨), Lθ

)

which, up to signs, intertwines the differentials, and hence induces the desired isomorphism of

complexes.

3.2 Application of formality

In this section we compute the Hochschild and cyclic (co)homology of the universal enveloping

algebra U(A) of a smooth Lie algebroid A over a manifold M. We define the Hochschild and

cyclic (co)homology taking into account the natural topology on U(A). The natural filtration

on the universal enveloping algebra leads to a spectral sequence that, in principle, allows one

to compute the Hochschild and cyclic (co)homology. However, it is not a priori clear that it

degenerates. As discussed in 1.2.6 Kassel circumvented this problem in [Ka] by constructing a

morphism of complexes

j : (Ω•S(g), Lθ) −→ (C•(A), b) j(Pdx1 · · ·dxn) =

∑σ∈Sn

(−1)|σ|η(P)xσ(1) ⊗ · · · ⊗ xσ(n)

where η is the symmetrization map Sym(g) → U(g), which induces an isomorphism in homology.

It requires a more sophisticated approach to construct a direct morphism in the general case

of smooth Lie algebroids, which relies on the formality map for Lie algebroids, c.f. [D, C].

Let us briefly explain this. Given a smooth Lie algebroid A → M, we showed in the previous

subsection that (DerR(SymA), SymA) =: (L, R) forms a sheaf of Lie algebroids over M. In

fact, L is triangular, i.e., there exists a Maurer–Cartan element θ ∈ Λ2RL = TL,1

poly(A∨), which is

given by the L-Poisson tensor. In the appendix, to which we refer if notations are unclear, we

use the PBW theorem for Lie algebroids to revisit the proof for the formality theorem for Lie

algebroids, which says that there exists an L∞-quasi-isomorphism U = (U1, U2, . . .)

U : TLpoly(A

∨)−→DLpoly(A

∨)

with the first component equal to the HKR map. The RHS is given by the L-polydifferential

operators on R, which are the polydifferential operators on A∨ that are polynomial along the

fibers, viewed as a sheaf over M. This morphism equips CL(A∨), the complex of L-Hochschild

chains, with the structure of an L∞-module over TLpoly(A

∨), and there exists a quasi-isomorphism

V = (V1, V2, . . .) of L∞-modules

V : CL(A∨) −→ ΩL(A

∨)

over TLpoly(A

∨), with V1 equal to the HKR map for chains. It is well-known that Maurer–

Cartan elements in TLpoly(A

∨) can be used to define Maurer–Cartan elements in DLpoly(A

∨),

which in turn define an L-deformations of R. We prove that the formality map and θ define a

deformation Rh of R = SymA which is isomorphic to U(Ah), where Ah is (a formal version of)

the adiabatic Lie algebroid discussed in [NWX], and that the product is well-defined for h = 1.




This is an extension of the discussion for Lie algebras in [K, §8.3.1]. Then we show that the

formality maps twisted by θ and composed with the natural morphisms induced by the anchor

ρ of A gives morphisms

(TL,•poly(A

∨), [θ, ]) −→ C•(U(A),U(A), b)

(C•(U(A), b) −→ (Ω•L(A

∨), Lθ).

Using a spectral sequence argument it is then showed that these morphism induce isomorphisms

in (co)homology. Finally, since the formality map for chains intertwines the de Rham differential

d for ΩL(A∨) and the cyclic differential B for cyclic homology, we can show that

(CW• (U(A)), b+ u−1B) −→ (Ω•

L(A∨)W , Lθ + u−1d)

is a morphism of complexes, where W is a K[u−1]-module. This proves the result for the

(periodic) cyclic homology.

3.2.1 Deformation quantization

Recall that a formal deformation quantization of a commutative ring R is given by a K[[h]]-

linear, associative product on R[[h]] satisfying

1 r = r = r 1, for all r ∈ R.

Writing out such a -product in powers of h,

r1 r2 = r1r2 +∑k1

hk

k!Bk(r1, r2) (3.2.1)

we see that it is given by a formal series B :=∑

k1hkBk/k! ∈ hC2(R, R)[[h]]. It is well known

that associativity of the product is equivalent to B satisfying the Maurer–Cartan equation in

the (shifted) Hochschild cochain complex:

δB+1

2[B, B]G = 0.

We encounter formal deformations controlled by the Lie–Rinehart algebra L: this means that

we can find Bk ∈ U(L)⊗R U(L) such that ρ(Bk) = Bk where ρ denotes the canonical extension

of the anchor to a morphism of cochain complexes

ρ : DL,•poly(R)[[h]] −→ C•(R, R)[[h]]

ρ(D1 ⊗ · · · ⊗Dn+1)(r1 ⊗ · · · ⊗ rn+1) := D1(r1) · · ·Dn+1(rn+1),(3.2.2)

where DL,npoly(R) is the degree n component of the L-polydifferential operators as defined in the

appendix. Following [NeTs01], we shall call such deformations L-deformations.

In the remaining part of this section we set R = SymA and L = DerR(SymA) for a smooth

Lie algebroid A. Recall the so-called adiabatic Lie algebroid Ah [NWX]. As a Lie–Rinehart

algebra, it is given by (Γ(A)[[h]], C∞(M)[[h]] with Lie bracket h[ , ] and anchor hρ, but it can

also be considered as a sheaf of Lie algebroids over M.

Proposition 3.2.1. Let A → M be a smooth Lie algebroid with associated triangular Lie

algebroid (L, R, θ). The formality L∞-morphism for L gives rise to a quantization Rh of R which

is isomorphic to U(Ah). Moreover, this isomorphism restricts to the subalgebras which are

polynomial in h.




Proof. Let U : TLpoly(A

∨) DLpoly(A

∨) be the L∞-morphism defined in the appendix. The

Maurer–Cartan element θ defines a Maurer–Cartan element in U(L)⊗RU(L)[[h]] by the formula:

∑k1

hk

k!Uk(θ, . . . , θ). (3.2.3)

This follows from the general theory for L∞-algebras as discussed in the appendix. Moreover,∑k1

hk/k!Bk, where Bk = Uk(θ, . . . , θ), defines a deformation quantization on R[[h]] by

(3.2.1). We will denote this deformed algebra by Rh.

The Maurer–Cartan element θ has polynomial degree −1 and we claim that the bidifferential

operators Bk have polynomial degree −k. This fact follows from the following observations.

Recall that the polynomial degree is detected by the formula

LE(f) = nf, f ∈ Symn A, E = Euler vector field on A∨.

The action of the Euler vector field can naturally extended to all the sheaves that are involved

in the proof of the formality morphism in the appendix. To show that the formality map

preserves the polynomial degree, it suffices to note that all the morphism that are used to

define the formality map respect the action of LE. For the natural resolutions using the jet

bundle this is clear and for the PBW map it follows because we can choose the L-connection

on L, c.f. (A.1.1) to be homogeneous :

LE(∇XY) = ∇LEXY +∇X(LEY), for all X, Y ∈ L.

Finally, for the fact that the twisted map preservers the degree we refer to the argument given

in [D] (there the property is described in terms of an action of a smooth Lie group, which is in

this case just the R∗ action on the fibers of A∨).

Concretely, this means that Bk is bidifferential operator mapping polynomials of degree p

and q to a polynomial of degree p + q − k, which in turn implies that we can restrict the

deformed product to the subalgebra of elements which are polynomial in h.

Now recall that the formal version of the adiabatic Lie algebroid is defined as the sheaf of

Lie algbroids A[[h]] over the ringed space C∞(M)[[h]]) with bracket and anchor defined by

[s, t]A[[h]] := h[s, t], ρA[[h]](s)(f) := hρ(s)(f).

We denote it by (Ah, C∞(M)h). Its universal enveloping algebra U(Ah) is, as an R[[h]-module,

isomorphic to Sym(A)[[h]]), and one can easily see that it is a deformation of the Poisson

algebra (Sym(A), , ). Because U(Ah) and Rh are deformations of (Sym(A), , ), i.e. of the

same Poisson bracket, they are, by an argument of Kontsevich in [K], isomorphic als algebras.

We can restrict the product in both deformations to the polynomial subalgebra, hence the

claim is proved.

Remark 3.2.2. The core of the proof above, the homogeneity of the bidifferential operators Bk,

shows that the quantization of the Poisson manifold A∨ defined by applying the formality map

for the Lie–Rinehart algebra (L, R) to the Lie–Poisson structure, is homogeneous as considered

in [NeWa]. For our purposes, it also shows that we can put h = 1 and consider R itself with the

deformed product defined by B. This is then isomorphic to the universal enveloping algebra

U(A) itself.

3.2.2 The basic complexes

Here we briefly recall the definitions of the (co)chain complexes computing the Hochschild and

cyclic theories of the universal enveloping algebras U(A) of a smooth Lie algebroid A. The




algebra U(A) has a natural locally convex topology defined as follows: we write U(A) as the

direct limit of the increasing family of filtered subspaces Fk ⊂ U(A), c.f. (2.1.2). By the

PBW theorem 2.2.4, Fk is the space of smooth sections of the vector bundle⊕k

i=0 Symi(A). It

therefore has a natural locally convex topology given by seminorms of the form

||f||p,K := supx∈K

X1,...,Xp

||∇X1· · · ∇Xp

f||Symk(A)X, K ⊂ M compact,

where we have chosen a connection ∇ and a metric on A. With respect to this topology, the

inclusions Fk ⊂ Fk+1 are continuous and we give U(A) the direct limit topology. As such, U(A)

is a nuclear LF-space, as each of the pieces Fk is nuclear Frechet. c.f. [Tr] for definitions. By

Proposition 3.2.1, the product fg of f, g ∈ Fk involves a finite number -at most k- of derivatives

of f and g. For each p ∈ N and K ⊂ M compact, we can find p ′ = p+ k such that

||f g||p,K C||f||p′,K||g||p′,K, for some C > 0.

The product : U(A) × U(A) → U(A) is therefore continuous for the topology and extends

to a map U(A)⊗U(A) → U(A), where ⊗ denotes the completed projective tensor product, so

U(A) is a locally convex algebra as in [Co][Appendix B].

The (topological) Hochschild cohomology is defined as the cohomology of the complex de-

fined on the graded vector space

C•(U(A),U(A)) := Homcts

(U(A)⊗•,U(A)

),

with differential δ := [ , ]. The dual Hochschild chain complex is given by

C•(U(A)) := U(A)⊗(•+1),

with differential written as b. As U(A) is a unital algebra, the B-operator defining the cyclic

homology of U(A) is given by the usual formula.

3.2.3 Computation of the cyclic theory

In this section we use the previous results to compute the topological Hochschild cohomology

of U(A). The final result is:

Theorem 3.2.3. Let A → M be a smooth Lie algebroid over a manifold M. There is a

canonical isomorphism

H•(U(A)) ∼= H•(A, Sym(ad A)).

Proof. The tangent map of the formality morphism, c.f. A.2 associated to the Maurer–Cartan

element θ ∈ TL,1poly(A

∨) is given by

TθU :(TLpoly(A

∨), [θ, ])−→

(DL

poly(A∨)[h], δ+ [θ, ]

),

where we could restrict to polynomials in h because of proposition 3.2.1. In fact, this morphism

is an L∞-moprhism, but we don’t need the Lie algebra structures for this proof. The same

definition as the natural morphism DL,•poly(A

∨)[h] −→ C•(R, R)[h] from (3.2.2) gives a morphism

of complexes (DL

poly(A∨)[h], δ+ [θ, ]

)−→

(C(Rpol

h , Rpolh ), δh

). (3.2.4)

This is not entirely obvious, but it follows from a rather straightforward computation relating

the differentials δ+ [θ, ] and δh -the Hochschild differential for the deformed algebra Rpolh . We

set h equal to 1, and use proposition 3.2.1 to obtain a morphism of complexes

(TLpoly(A

∨), [θ, ])−→ (C•(U(A),U(A)), δ) .




Let us now consider the filtration on the Hochschild complex induced by the filtration on U(A):

first define

Fp(U(A)⊗k) =∑

i1+...ik=p

Fi1(U(A))⊗ . . .⊗ Fik(U(A)).

And with this we put

Fp(Ck(U(A),U(A)) =

∑l

Homcts

(Fl(U(A)⊗k), Fl+p(U(A))

).

This induces a spectral sequence with

Ep,q0 = FpC

p+q/Fp−1Cp+q ∼= Homp

cts

(R⊗(p+q), R

).

The differential d0 : Ep,q0 → Ep,q+1

0 is just the standard Hochschild differential of the commu-

tative topological algebra R. By the topological version of the Hochschild–Kostant–Rosenberg

theorem we therefore find

Ep,q1

∼= TLpoly(A

∨).

The differential on this page of the spectral sequence is defined using the first term B1 and is, by

the same reasoning as in [Br], equal to the Poisson cohomology differential [θ, ]. If we now view

(TLpoly(A

∨), [θ, ]) as a filtered complex on its own, with the filtration by polynomial degree,

we can see that TθU is a morphism of filtered complexes as follows: The formality morphism

preserves the polynomial degree, and the tangent map

TθU = U1 + terms containing θ

where U1 is the HKR map which is of degree 0, and the terms containing θ strictly decrease the

degree because θ has degree −1. Because of this, we see that the induced map on the spectral

sequence is an isomorphism from the first page on. By the comparison theorem for spectral

sequences, we conclude that TθU is a quasi-isomorphism itself.

Finally, the theorem follows from the identification of the polynomial Poisson cohomology

with the Lie algebroid cohomology of symmetric powers of the adjoint, c.f. Theorem 3.1.14.

Theorem 3.2.4. Let A be a smooth Lie algebroid over a manifold M. There exists an isomor-

phism

H•((U(A),U(A)) ∼= H•θ

(Ω•

L(A∨)

)

where H•θ(Ω

•L(A

∨)) is the Poisson homology of the complex (Ω•L(A

∨), Lθ).

Proof. This follows by an argument which is very similar to the proof of the last theorem, but

here one uses the globalized and twisted version of Shoikhets formality map:

(CL(A

∨), bH + Lθ) Vθ

−→(Ω•

L(A∨), Lθ

). (3.2.5)

Moreover, we use the inclusion (C•(U(A),U(A)), bH) −→(CL(A

∨), bH + Lθ)which is the

composition of the natural injective map

α1 : C•(R, R) −→ C•(J(L), J(L))

and the projection (A.4.2). The description of the first page of the spectral sequence associated

to the filtrated Hochschild complex of U(A) can be found in [Br].

Theorem 3.2.5. Let A be a smooth Lie algebroid over a manifold M. There exists a quasi-

isomorphism of complexes

(C•(U(A)), b+ u−1B

)−→

(Ω•

L(A∨), Lθ + u−1d

).




Proof. This is done in a similar fashion, but now the differentials B and d are involved. We refer

to lemma A.6.1 for the fact that the twisted formality map of modules in (3.2.5) intertwines

the differentials B and d. Moreover, in [BrGe, Wo], the authors proved that the differential on

the second page of the spectral sequence of the cyclic complex of U(A) coincides with the de

Rham differential d. There are no further difficulties.

For the periodic theory, the homology simplifies:

Corollary 3.2.6. For a smooth Lie algebroid A → M we have

HP•(U(A)) ∼= H•(M)((u−1)).

Proof. Recall that the periodic cyclic homology is computed from the complex C•(U(A))((u−1))

with differential b+u−1B. The quasi-isomorphism of the previous theorem extends to a quasi-

isomorphism to the complex Ω•L(A

∨)((u−1)) equipped with the differential Lθ + u−1d. The

invertible map

euιθ : Ω•L(A

∨)((u−1)) → Ω•L(A

∨)((u−1))

simplifies the differential to just d. But the cohomology of the complex Ω•L(A

∨)((u)) is easy to

compute: Any homogeneous element α ∈ Ω•L(A

∨)((u−1)) of degree p is an eigenvector of the

Euler vector field: LEα = pα. But this implies that when α is d-closed, it is also exact: α =

p−1dιEα. This argument shows that the inclusion π∗ : Ω•(M)((u−1)) → Ω•L(A

∨)((u−1)) -the

pull-back along the projection map π, is a quasi-isomorphism. This proves the statement.

3.2.4 Some examples

Trivial Lie algebroids

For the trivial Lie algebroid A = 0×M, the universal enveloping algebra reduces to the algebra

of smooth functions C∞(M). In this case the adjoint representation has only one nonzero

component in degree 1 where it is just TM, so Sym(ad A) = Λ•TM with zero differential. The

Lie algebroid DerR(SymA) is equal to TM and the Poisson structure θ ∈ Λ2L is given by the

trivial one, hence [θ, ] = Lθ = 0. Clearly, we have

(Ω•L, Lθ + dL) = (Ω(M), d),

so in this case our computations reduce to the well-known isomorphisms of Connes [Co]:

HH•(C∞(M)) ∼= Λ•TM

HH•(C∞(M)) ∼= Ω•(M)

HC•(C∞(M)) ∼= H•(Ω(M)[u−1], u−1d)

HP•(C∞(M)) ∼=

⊕k0

H•+2kdR (M).

Lie algebras

A Lie algebroid over a point is the same as an ordinary Lie algebra g. In this case the adjoint

representation exists on g as an honest representation. The isomorphism between the Poisson

cohomology complex of the Poisson algebra Symg and the Lie algebra cohomology complex

with values in the symmetric algebra of the adjoint representation is given by:

(TL,•poly(A

∨), [θ, ]) ∼= (C•(g,Sym g), dLie).




Here L = Sym g ⊗ g∗. It has a dual counterpart, namely an isomorphism between the Poisson

homology complex of the Poisson algebra Sym g and the Lie algebra homology complex with

values in the symmetric algebra of the adjoint representation:

(Ω•L, Lθ)

∼= (C•(g,Sym g), dLie).

Our computations of the (co)homology of the universal enveloping algebra U(g) coincide with

the results from Kassel [Ka]

H•(U(g)) ∼= H•Lie(g; Sym(g))

H•(U(g)) ∼= HLie• (g; Sym(g))

HP•(U(g)) ∼= C((u−1))

HC•(U(g)) ∼= H•(Ω•

L(g∗)[u−1], Lθ + u−1d

).

Tangent spaces

Consider the Lie algebroid A = TM. The universal enveloping algebra of TM is given by the

differential operators Diff(M) on M. The Lie algebroid L = DerR(Sym TM) has a Poisson

structure which is non degenerate, and this simplifies the discussion. We have the following

lemma:

Lemma 3.2.7. The maps i and π

(Ω(M), d) (ΩL, dL) (ΩL, dL) (Ω(M), d)i π

given by the natural injection and projection are homotopy inverses with respect to the given

differentials. Hence

H•dR(M) ∼= H•

Lie(L)

Proof. This a copy of the proof of proposition 4.5 from [L-BM].

Brylinski proved in [Br] that the ’hodge star’ map for a symplectic manifold gives an iso-

morphism of the de Rham complex with the Poisson homology complex:

(Ω•L, dL) −→ (Ω2n−•

L , Lθ).

Since the modular class of a symplectic manifold vanishes, the Poisson homology complex and

the Poisson cohomology complex are isomorphic as well:

(Ω2n−•L , Lθ) ∼= (TL,•

poly, [θ, ·]),

so we conclude that H•dR(M) ∼= H•(A, Sym(adA)) and H•

dR(M) ∼= Hθ2n−•(SymA), where the

last notation means the Poisson homology of the sheaf of Poisson algebras SymA over M.

Hence, in this case our computations reduce to the results of Wodzicki [Wo]:

HH•(Diff(M)) ∼= H•dR(M)

HH•(Diff(M)) ∼= H2n−•dR (M)

HC•(Diff(M)) ∼= H2n−qdR (M)⊕H2n−q+2i

dR (M)⊕ . . .

HP•(Diff(M)) ∼= Hev/odddR (M).






Chapter 4

The trace density map

In this chapter we construct a trace density map for the universal enveloping algebra of a Lie

algebroid. We assume that L is either a smooth Lie algebroid or a holomorphic Lie algebroid,

but we think that most of the constructions can be done in greater generality. In the previous

chapter we abstractly identified the cyclic theory of the universal enveloping algebroid U(A)

of a smooth Lie algebroid; the result was that it is given by the cohomology of the symmetric

algebra of the adjoint representation (twisted by the bundle QA). The trace density map, which

is extensively studied in index theory, gives a more explicit construction.

The trace density in deformation quantization made its first appearance in the fundamental

papers [NeTs95, NeTs96, BNT] on the algebraic index theorem. There, the authors used

Cech methods and an abstract description of the cyclic cohomology of the Weyl algebra to

prove an index theorem. In [FFS] the authors constructed an explicit cocycle in the Hochschild

cohomology of the Weyl algebra with invariance properties and used it to define a map from the

Hochschild homology of a deformation of the smooth functions on a symplectic manifold to de

Rham forms of top degree of that manifold. The canonical trace associated to the deformation

was then defined as the integral of the image of smooth functions with compact support over

M. Using the local Riemann–Roch theorem, c.f. [FT, FFS], which identifies the image of the

Hochschild cocycle under a natural map to the Lie algebra cohomology as certain characteristic

classes, the authors proved that the trace applied to the canonical class of 1 in the Hochschild

homology is equal to the integral of certain characteristic classes, i.e. topological data. In

[EnFe], the Hochschild cocycle was used to define a morphism from the Hochschild homology of

holomorphic differential operators on a complex vector bundle E → X to the sheaf of smooth de

Rham forms on X, and they used it to prove that the supertrace of a holomorphic differential

operator can be expressed as an integral over topological data. In [PoPfTa10] and [W15] the

fundamental Hochschild cocycle was extended to a cyclic cocycle of the Weyl algebra, and

the authors of [PoPfTa10] used it to give a proof of the algebraic index theorem of [NeTs95].

Roughly speaking, given a deformation of the smooth functions on a symplectic manifold, the

algebraic index theorem computes the pairing of the Chern character of an element in the K0

theory of the deformation with a smooth de Rham form as an integral over topological data.

These ideas were used in [PoPfTa10] to prove the higher index theorem of [CoMo]. Generalizing

to Lie groupoids, in [PoPfTa15] the authors applied similar techniques to construct a pairing

for elliptic elements of the universal enveloping algebroid of a smooth, integrable Lie algebroid.

This pairing was expressed as an integral over A∗. The image of our trace density map is given

by the L-valued forms, and it would be interesting to investigate whether similar pairings can

be constructed for our trace density map.

The trace density map that we construct is a morphism from the sheaf complex of cyclic

chains of the universal enveloping algebra to a Cech resolution of the de Rham complex of L-

71



72 CHAPTER 4. THE TRACE DENSITY MAP

forms. When L is smooth, the construction can be performed globally, providing a morphism

from the cyclic complex of (global sections of) the universal enveloping algebra to the complex

of global section of L-forms.

Now we outline our construction. Our trace density map depends on the PBW isomorphism,

and hence on the choice of a connection, which explains that it can be globally defined when L is

smooth. We show that the associated map in (co)homology does not depend on the choice of a

connection in the smooth case. When L is holomorphic, global connections do not always exist.

However, we can choose an open cover U such that L is locally trivial over each Ui ∈ U , which

implies that we can choose local connections and we obtain a morphism of sheaves (restricted

to Ui)

ΦUi: CC•(U(L))|Ui

−→ Tot2r−•ΩL|Ui.

Here Tot•ΩL :=⊕

i0 Ω•−2iL . Over intersections Ui0,...ik , we consider an affine connection∇aff

depending on a variable t of the k-simplex, and the associated family of PBWisomorphisms.

We use the derivatives of this family, c.f. 2.2.10, to define maps

ΦUi0,...,ik: CC•(U(L))|Ui0,...,ik

−→ Tot2r−•ΩL|Ui0,...,ik

This implies that, given a section D of the sheaf CC•(U(L)), we can send it to the Cech

resolution of Ω•L with respect to U , and we obtain a morphism

ΦU : CC•(U(L)) −→ C2r−•(U ,TotΩL).

We will show that this morphism intertwines the differentials b + u−1B on the LHS and d +

(−1)kδ on the RHS, so in fact it is a morphism of complexes of sheaves. Moreover, by the same

reasoning as in the smooth case, this morphism does not depend on the choice of connections

on Ui. The hypercohomology of Ω•L coincides with the hypercohomology of C(U ,Ω•

L) for any

cover of X and we show that the trace density map on the level of hypercohomology

Φ : H•(CC•(U(L)), b+ u−1B) −→ H•(TotΩL,W , dL)

does not depend on the choice of a cover and local connections.

In section 4.2 we discuss characteristic classes in the setting of Cech cohomology, in order

to compare them with the image of our trace density map. This construction is a particular

example of the characteristic classes for simplicial manifolds, c.f. [Du]. Then we identify the

image of the canonical class of 1 in the cyclic homology under the trace density map in the

twisted case, i.e., we consider the universal enveloping algebra with coefficients in a vector

bundle E. It is given by the product of the Todd class of L and the L-valued Chern character

of the vector bundle E. Note that we complexify the bundle in the case where L and E are

smooth to define the Todd class. Finally, we show that the trace density map is compatible

with the HKR map.

4.1 Definition and properties of the trace density map

4.1.1 The set up

We assume that (X,OX) is either a smooth manifold with the sheaf of smooth functions, or a

holomorphic manifold with the sheaf of holomorphic functions. Moreover, L is either a smooth

or a holomorphic Lie algebroid of rank r over (X,OX). We define the sheaf complex of the cyclic

homology of OX as the sheaf associated to the following presheaf:

U → (CCW• (OX(U)), b+ u−1B), U ⊂ X open.



4.1. DEFINITION AND PROPERTIES OF THE TRACE DENSITY MAP 73

Recall that u−1 is a formal variable of degree −2 (so that b+u−1B has degree −1) and that we

write CCW• (OX(U)) = CC•(OX(U))[[u−1]] ⊗K[u−1] W where W is one of the K[u−1]-modules

discussed in the section about cyclic theory. The usual Hochschild–Kostant–Rosenberg map

HKR : f0 ⊗ . . .⊗ fk → f0df1 ∧ . . .∧ dfk, fi ∈ OX(U), i = 0, . . . , k, (4.1.1)

sheafifies to define a morphism (CCW• (OX), b, u

−1B) → (Ω•OX,W , 0, u−1d) of mixed complexes

of sheaves, where Ω•OX,W = Ω•

OX⊗K[u−1] W.

Remark 4.1.1. When X is a manifold, it is customary to complete the sheaf of cyclic chains

by taking the germs or jets of functions around the diagonal in X×(k+1). We do not consider

such completions here, neither are we concerned with the question whether the HKR map is

a quasi-isomorphism. However, it is not difficult to prove that in this case our index theorem

extends to the completed chain complexes.

We define the cyclic chain complex of sheaves associated to the sheaf of universal enveloping

algebras for a given Lie algebroid L in a similar fashion as the sheaf associated to the following

presheaf of complexes:

U → (CCW• (U(L(U))), b+ u−1B).

The associated sheaf of chain complexes is simply denoted by (CC•(U(L)), b+ u−1B).

The aim of this section is to construct a map between this complex of sheaves and the de

Rham complex of sheaves (Ω•L, dL). It is the analogue of the so-called “trace density” [BNT]

or “character map” [PoPfTa10, Theorem 3.9] in deformation quantization. In our setting, the

proper formulation is in terms of the derived category D(X) of K-sheaves on X. To state the

precise theorem, we define, Ω•L,W := Ω•

L[[u]−1] ⊗K[u−1] W, for any K[u−1]-module W. In the

language of derived categories, the main theorem of this section then reads as follows:

Theorem 4.1.2. Let (L,OX) be a Lie algebroid. There exists a canonical morphism in the

derived category of K-sheaves:

Φ ∈ HomD(X)

((CCW

• (U(L)), b+ u−1B),(Tot2r−•ΩL,W , dL

)).

Recall that the derived category of K-sheaves is constructed out of the category of chain

complexes of sheaves by 1) identifying homotopic chain maps (i.e., going over to the homotopy

category and 2) localization with respect to quasi-isomorphisms. For any cover U of X, the nat-

ural map from the Lie algebroid chain complex (Ω•L,W , dL) to its Cech resolution C(U ,Ω•

L,W)

is a quasi-isomorphism, hence a morphism

ΦU :(CCW

• (U(L)), b+ u−1B)−→ C•−2r(U ,TotΩL,W , dL + (−1)2r−•δ)

determines a morphism in the derived category:

[ΦU ] ∈ HomD(X)

((CCW

• (U(L)), b+ u−1B),(Tot2r−•ΩL,W , dL

)).

The proof of the theorem that we give consists of three steps: 1) The construction of a morphism

ΦU for any open cover U such that L is trivial over all Ui ∈ U ). 2) We show that ΦU

intertwines the differential of the complexes of sheaves. 3) We show that the morphism in the

derived category does not depend on the choices that were made.

When L is smooth, it implies that there exists a canonical morphism on the cohomology

and when L is holomorphic it implies that there exists a canonical map in hypercohomology.

4.1.2 The Fedosov resolution

In this subsection we assume that L admits a global L-connection ∇. We construct the ana-

logue of the Fedosov resolution [Fed] for U(L). In the smooth category, such a resolution was




constructed in [NeWa] from the point of view of formal deformation quantization of the dual

of a Lie algebroid equipped with the Lie–Poisson bracket. Here we use the PBW theorem of

chapter 2. Recall from this chapter that

α2 : U(L) −→ J(L)⊗OX2U(L) ∼= DiffOX1

(J(L)),

where the first map is given by D → α2(1)⊗D and the second map is defined using the second

U(L)-module structure on J(L). There is a natural product on the RHS given by composition

of differential operators. The flat connection ∇(1) on J(L) induces in a natural way a flat

connection on DiffOX1(J(L)), which is given by ∇(1) ⊗ 1 on J(L) ⊗OX2

U(L). We denote the

associated Chevalley–Eilenberg complex by

Ω•L(DiffOX1

(J(L))) :=(J(L)⊗OX2

U(L))⊗OX1

Λ•OX

L∨

and the differential by d∇(1) .

Lemma 4.1.3. The Chevalley–Eilenberg complex(Ω•

L

(DiffOX1

(J(L))), d∇(1)

)is a sheaf of

differential graded algebras.

Proof. By construction, Ω•L

(DiffOX1

(J(L)))is both a graded algebra (combining the multipli-

cation with the wedge product) and a cochain complex. Therefore, the only thing to check is

the compatibility between both structures. The action of ∇(1) on DiffOX1(J(L)) is defined by

∇(1)X (D)(φ) := [∇(1)

X , D](φ), X ∈ L, D ∈ DiffOX1(J(L)) (4.1.2)

and hence it follows easily that ∇(1) acts by derivations on DiffOX1(J(L)), which proves the

lemma.

Proposition 4.1.4. The map U(L) → Ω•L(DiffOX1

(J(L))) is a quasi-isomorphism of complexes

of sheaves.

Proof. The proof is very similar to the proof of proposition 4.2.4 in [CvdB]. The adic filtration

on J(L), c.f chapter 2, induces a natural filtration on the sequence:

U(L)α2−→ J(L)⊗OX2

U(L)d∇(1)−→ J(L)⊗OX2

U(L)⊗OX1Λ1

OXL∨

d∇(1)−→ . . .

Note that L∨ has degree 1. The associated graded exact sequence is given by

U(L)i−→ SymOX

(L∨)⊗OXU(L)

−dK−→ SymOXL∨ ⊗OX

U(L)⊗OXΛ1

OXL∨ −dK−→ . . .

where −dK is (minus) the standard Koszul differential on SymOX(L∨) ⊗OX

ΛOXL∨ given by∑

(ιli ⊗ l∗i ) in local coordinates li on L, and the identity on U(L). Since −dK is exact, the

original sequence is exact.

Remark 4.1.5. In the smooth category, it is a well-known fact that the universal enveloping

algebra of a Lie algebroid is isomorphic to the algebra of invariant differential operators on an

integrating Lie groupoid. The preceeding Proposition should be viewed as an algebraic counter-

part of that statement, with the integrating Lie groupoid being replaced by the Hopf algebroid

of jets, and instead of taking invariants, we consider the cohomology of Ω•L1

(DiffOX1(J(L))).

The PBW theorem, which is a OX1-linear algebra isomorphism, induces the following natural

isomorphisms:

(ΩL(J(L)), d∇(1)) −→(ΩL(SymOX

(L∨)), d∇(1)

)(ΩL(DiffOX1

(J(L))), d∇(1)

)−→

(ΩL(DiffOX

(SymOX(L∨))), d∇(1)

)




where d∇(1) = j∇ d∇(1) j−1∇ . This equips the right hand sides with a dg algebra structure.

The L-connection ∇ induces natural derivations of degree 1 on the algebras ΩL(SymOX(L∨))

and ΩL(DiffOX(SymOX

(L∨))). These derivations, which we denote by ∇ as well, do not square

to 0 unless ∇ happened to be flat. Let us explicitly compare the operators ∇ and d∇(1) on

SymOX(L∨):

A(∇) := j∇ dnabla(1) j−1∇ −∇. (4.1.3)

Since d∇(1) and ∇ act by derivations, it follows that A(∇) ∈ Ω1L(Der(SymOX

(L∨))). The

difference of the two operators ∇ and d∇(1) on the graded algebra DiffOX(SymOX

(L∨))) is

given by

ad(A(∇)) = [A(∇), ] = j∇ d∇(1) j−1∇ −∇.

This follows from the fact that ∇ and d∇(1) are defined as in (4.1.2). The crucial property of

A(∇) is that it satisfies the Maurer–Cartan equation, which is the content of the next lemma.

Lemma 4.1.6. Let A(∇) be as defined in (4.1.3). The following formula holds:

R(∇) +∇A(∇) +1

2[A(∇), A(∇)] = 0. (4.1.4)

Proof. This follows from the fact that∇+A(∇) is a differential. Note that R(∇) is the curvature

of ∇.

4.1.3 A proof of Theorem 4.1.2

As explained after theorem 4.1.2, we will define the trace density map as a morphism from the

sheaf complex of the cyclic complex of U(L) to a Cech resolution of the comples of L-forms.

Then we show that the induced morphism in the derived category does not depend on the

choices that are made. Since we apply the PBW theorem, we choose a cover U = Uii∈I

of X by opens Ui such that L|Uiis trivial for each Ui ∈ U . This implies that there exist

L|Ui-connections on L|Ui

. A concise way to formulate the Cech complex is provided by the

notion of the Cech groupoid:

XU :=∐i,j∈I

Ui ∩Uj ⇒∐i∈I

Ui.

For each intersection in the k-th nerve XU ,k :=∐

i0,...,ik∈I Ui0 ∩ . . .∩Uik , we define the family

of connections

∇affUi0,...,ik

:=

k∑j=0

tij∇ij , where (ti0 . . . tik) ∈ ∆k = (ti0 . . . tik) ∈ Rk+1 :

k∑j=0

tij = 1

In the following, we simply write t = (ti0 , . . . , tik) ∈ ∆k. To this “simplicial connection”

∇affUi0,...,ik

we can associate the following elements:

1. The family of connection forms AaffUi0,...ik

(t) as in formula (4.1.3).

2. The derivations θUi0,...,ik

j (t), j = 1, . . . , k by applying propostion 2.2.10 to the family of

PBW isomorphisms induced by the affine connection. To be precise, θUi0,...,ik

j (t) is the

t-dependent derivation of SymOX(L∨) that satisfies

θUi0,...,ik

j (t) j∇aff = (∂tij − ∂tij−1)j∇aff (t). (4.1.5)

These elements, and the cyclic cocycle τ2r in the cohomology complex of the Weyl algebra, c.f.

B, appear in the definition of the trace density map. Let D = D0 ⊗ . . .⊗Dl ∈ CCWl (U(L)).




Definition 4.1.7. Given a cover U such that L is locally trivial over each Ui ∈ U , the trace

density map

ΦU : CC•(U(L)) → C2r−•(U ,TotΩL)

is defined by the following equation:

ΦUi0...ik(D) := (−1)α

∫

∆k

ιθ1. . . ιθk

τ2r(1⊗ exp(∧Aaff(t))×D)dt. (4.1.6)

where Ui0,...ik ⊂ XU ,k, the k-th Cech nerve of the cover U .

Let us clarify this definition.

1. We suppressed a number of super and subscripts that we used before to keep the notation

readable.

2. The symbol dt indicates the natural volume form on the simplex induced by the standard

volume form on Rn.

3. The symbol D stands for j∇aff (D) ∈ DiffOX(SymOX

(L∨)).

4. The cyclic cocycle τ2r is a cocycle for the Weyl algebra W(L), thus, to apply it to elements

in DiffOX(SymOX

(L∨)), we use that DiffOX(SymOX

(L∨)) ∼= W(L).

5. The product is given by the graded shuffle product of elements inΩL(W(L)). The grading

is given by the degree of differential forms.

6. The element exp(∧A) is an L-form with values in the Weyl algebra such that

exp(∧A)(X1, . . . , Xn) = A(X1)∧ . . .∧A(Xn)

7. Finally, the sign is given by α =∑k

i=1 i+∑l

j=1 j.

This was step 1 of the proof of theorem 4.1.2; the following proposition deals with step 2.

Proposition 4.1.8. The map ΦU defined in equation (4.1.6) is a morphism of complexes:

ΦU :(CCW

• (U(L)), b+ u−1B)−→

(C2r−•(U ,TotΩL,W , d+ (−1)kδ

).

Let us simply write jaff for the family of PBW isomorphisms j∇aff induced by the connection

∇affk parametrized by t ∈ ∆k. We parametrize the k-simplex ∆k by (t1, . . . , tk) ∈ Rn : tj

0,∑

tj 1 which allows to write ∂tj instead of ∂tj − ∂tj−1, c.f. (4.1.5). The following Lemma

is an easy consequence of proposition 2.2.10.

Lemma 4.1.9. The following equations hold true:

∂ti jaff = adθi

(jaff)

∂tiAaff = adθi

(Aaff) −∇affθi − ∂ti∇aff

∂tiθj − ∂tjθi = [θi, θj].

Proof. This follows from straightforward computations. Let K ∈ DiffOX(SymOX

(L∨). We

compute:

∂ti(jaff(K)) =∂ti(j

aff K (jaff)−1)

= adθijaff(K),




which is the first equality. The second equality follows from the identities

∂ti(Aaff +∇aff) =∂ti(j

aff d∇(1) (jaff)−1)

= adθi(jaff d∇(1) (jaff)−1)

= adθi(Aaff) −∇affθi.

Finally, the chain rule implies that

∂ti∂tj jaffk = (∂tiθj + θjθi)j

aff

which gives, together with the same equality where i and j are interchanged, the last expression.

Next, we need some equalities involving the Maurer–Cartan element A(∇) in the sheaf of

unital, graded algebras Ω•(W(L)). We set (A)k = 1 ⊗ A ⊗ . . . ⊗ A. Consult section 1.2.2 for

the definitions of cyclic homology and the graded shuffle product.

Lemma 4.1.10. For all a = a0 ⊗ · · · ⊗ ap ∈ Cp(Ω•(W(L))) the b operator satisfies the

following equalities:

b((A)k × a) =b((A)k)× a+ (−1)k(A)k × b(a)

+(−1)k∑i

(A)k−1 × (a0 ⊗ . . .⊗ [A,ai]⊗ . . . al),

b((A)k) =∇((A)k−1).

The cyclic differential B satisfies the following relation:

B((A)k × a) =(−1)k(A)k × B(a).

This lemma holds for any dg algebra with a Maurer–Cartan element.

Proof. Entirely analogous to [EnFe, Lemma 2.5 & 2.6] and [PoPfTa10, Lemma 3.3].

In the following remark we provide some details on the identification of the Weyl algebra

with the differential operators, which we will use in the proof.

Remark 4.1.11. Recall that the sheaf of Weyl algebras W(L) is constructed as follows. The

direct sum L⊕L∨ has a natural symplectic structure, and W(L) is the sheaf of Weyl algebras

with fibers W(L)|x given by the Weyl algebra associated to the symplectic vector space (L ⊕L∨)|x. There is a natural fiberwise identification

SymOX(L∨)⊗ SymOX

(L) ∼= W(L) ∼= DiffOX(SymOX

(L∨)).

We introduce local coordinates li on L and dual coordinates l∗i on L∨, moreover we denote the

coordinates on the symplectic vector spaces (L⊕L∨)|x by qi and pi. Under the identification

above we have

(l∗i ⊗ li)x → piqi +1

2, l∗i ⊗ li ∈ Sym

1

OX(L∨)⊗ Sym1

OX(L).

Proof of Proposition 4.1.8. It suffices to check the proposition on a fixed intersection Ui0,...,ik .

Unraveling the definitions of the trace density map, we need to show that

ΦUi0,...,ik(b+ u−1B)(D) = dL(ΦUi0,...,ik

(D)) + (−1)kδ(ΦUi0,...,ij,...ik(D)). (4.1.7)




] We denote the coordinates on the simplex ∆k by t1, . . . . . . tk, where we used the embedding

∆k ⊂ Rk. The coordinates on the simplices corresponding to the intersections Ui0,...ij,...ikare

denoted by t1, . . . , tj, . . . , tk for j = 0 and by t2, . . . tk for j = 0. Stokes’ theorem implies:

δ(ΦUi0,...,ij,...ik(D)) =

k∑j=0

(−1)jΦUi0,...,ij,...ik(D)

=

k∑j=0

(−1)j∫

∂j∆k

(k∑

i=1

ιθ1. . . ιθi

. . . ιθkτ2s(A×D)dt1 . . . dti . . . dtk

)

=

∫

∆k

d∆k

k∑j=1

ιθ1. . . ιθj

. . . ιθkτ2s(A×D)dt1 . . . dti . . . dtk

=

∫

∆k

k∑j=1

(−1)j+1∂tj(ιθ1

. . . ιθj. . . ιθk

τ2s(A×D))dt1 . . . dtk.

The symbol τ2s indicates the 2s component of the cyclic cocycle τ2r and we wrote A for (A)j

The number j can be deduced from the formula, but sometimes we will write it for clarity.

Hereafter we will be concerned with rewriting the integrand in the last expression. We omit

the term dt1 . . . dtk. First we split the integrand into two terms:

k∑j=1

(−1)j+1ιθ1. . . ιθj

. . . ιθk∂tj (τ2s(A×D)) (∗)

+∑j=i

(−1)j+1ιθ1. . . ι∂tj

θi. . . ιθj

. . . ιθk∂tjτ2s(A×D) (∗∗)

The relations for the derivatives of jaff and Aaff , c.f. lemma 4.1.9, imply that

(∗) =∑j

(−1)j+1ιθ1. . . ιθj

. . . ιθkτ2s(Lθj

(A×D))

+∑j

(−1)j+1∑i

ιθ1. . . ιθj

. . . ιθk(τ2s((1⊗A⊗ . . .⊗−∇θj︸︷︷︸

i

⊗ . . .⊗A)×D)

=∑j

(−1)j+1Lθjιθ1

. . . ιθj. . . ιθk

τ2s(A×D))

+∑j

(−1)j+1ι∇θjιθ1

. . . ιθj. . . ιθk

τ2s(A×D)),

where we omitted the terms involving ∂ti∇aff ∈ Ω1L(Sym

1

OX(L∨) ⊗ Sym1

OX(L). The reason is

the following: the identification of remark 4.1.11 shows that ∂ti∇aff corresponds to a constant

term and quadratic terms in the Weyl algebra. The definition of the cocycle shows that the

constant terms vanish, and the quadratic terms are zero because τ2r is basic, c.f. equation

(B.2.3). The relation ∂tiθj − ∂tjθi = [θi, θj] can be used to show that

(∗∗) =∑j=i

(−1)j+1ιθ1. . . ι∂tj

θi. . . ιθj

. . . ιθkτ2s(A×D)

=∑i<j

(−1)jιθ1. . . ι[θi,θj] . . . ιθj

. . . ιθkτ2s(A×D).

Moreover, we have an equality derived from [La, ιb] = ι[b,a]:

∑j

(−1)j+1ιθ1. . . Lθj

. . . ιθkτ2s(A×D) =

∑j

(−1)j+1Lθj. . . ιθj

. . . ιθkτ2s(A×D)

+∑i<j

(−1)jιθ1. . . ι[θi,θj] . . . ιθj

. . . ιθkτ2s(A×D).




Collecting the previous formulas amounts to

(∗) + (∗∗) =∑j

(−1)j+1ι∇θjιθ1

. . . ιθj. . . ιθk

τ2s((A)2s−k−l ×D))

+∑j

(−1)j+1ιθ1. . . Lθj

. . . ιθkτ2s((A)2s−k−l+1 ×D).

Applying the relation bιa + ιab = La several times gives

∑j

(−1)j+1ιθ1. . . Lθj

. . . ιθk= bιθ1

. . . ιθk+ (−1)k+1ιθ1

. . . ιθkb.

Moreover, we have the following two equalities, obtained from lemma 4.1.10 and a simple

relation between the shuffle operation to the insertion operator:

b((A)2s−k−l+1 ×D) =∇(A)2s−k−l ×D− (−1)k+l(A)2s−k−l+1 × b(D)

−(−1)k+l∑i

(A)2s−k−l ×D0 ⊗ . . .⊗ [A,Di]⊗ . . .⊗Dl

ιθ1. . . ιθk

bτ2s(A×D) =(−1)kτ2s+2(B((θk)× . . .× (θ1)×A×D))

=(−1)k+l+1τ2s+2((θk)× . . .× (θ1)×A× B(D))

=(1)l+1ιθ1. . . ιθk

τ2s+2(A× B(D)).

We know that D is a flat section for the connection ∇(1). This implies that ∇Di + [A,Di] = 0

and hence the following identies hold:

(∗) + (∗∗) =∑j

ιθ1. . . ι∇θj

. . . ιθkτ2s((A)2s−k−l ×D))

+ ιθ1. . . ιθk

(τ2s(∇(A)2s−k−l ×D) + (−1)k+lτ2s((A)2s−k−l ×∇(D))

)

+ (−1)k+l+1ιθ1. . . ιθk

τ2s((A)2s−k−l+1 × b(D))

+ (−1)k+l+1ιθ1. . . ιθk

τ2s+2((A)2s−k−l+1 × B(D))

=dL (ιθ1. . . ιθk

τ2s(A×D)) + (−1)k+l+1ιθ1. . . ιθk

τ2s(A× b(D))

+ (−1)k+lιθ1. . . ιθk

τ2s+2(A× B(D)),

which is equivalent to Equation (4.1.7). Of course, the sign α in Definition 4.1.6 has to be taken

into account.

We are ready to give the proof of the main theorem:

Proof of Theorem 4.1.2. Recall that the derived category of sheaves is constructed out of the

category of chain complexes of sheaves by 1) identifying homotopic chain maps (i.e., going

over to the homotopy category and 2) localization with respect to quasi-isomorphisms. For

any cover U , the natural map from the Lie algebroid chain complex (Ω•L,W , dL) to the Cech

resolution C•(U ,ΩL,W , dL + (−1)kδ) with respect to the cover is a quasi-isomorphism, so by

Proposition 4.1.8 the character map ΦU defines a morphism in the derived category as stated

in Theorem 4.1.2. It therefore remains to be shown that this map is independent of the cover

and the compatible connections chosen.

Suppose we are given two covers (U ,∇U ) and (V ,∇V ) and compatible families of connec-

tions. We can merge the two covers U and V into one cover U∐

V which consists of the

intersections Ui ∩ Vj for all Ui ∈ U and Vj ∈ V . We denote the cover U∐

V with all the




connections induced by the connections on U by (U∐

V ,∇U ), and we write (U∐

V ,∇V )

for the cover U∐

V with the connections induced by the connections on V . It is obvious that

the following diagram is commutative:

CC•(U(L))(C2r−•(U ,TotΩL, dL + (−1)kδ

)

(C2r−•(U

∐V ,TotΩL, dL + (−1)kδ

).

ΦU

ΦU∐

V ,∇U

Since the right arrow is a quasi-isomorphism, the morphisms ΦU and ΦU∐

V ,∇U induce

the same morphism in the derived category. A similar diagram shows the same for ΦV and

ΦU∐

V ,∇V , hence it suffices to show that ΦU∐

V ,∇U and ΦU∐

V ,∇V define the same mor-

phism in the derived category. Since the covers of these morphisms are the same we can assume

that the covers are trivial, i.e. we have prove that Φ∇1 and Φ∇2 are homotopic for ∇1 and ∇2

global L-connections on a space X. This follows if we can write the difference Φ∇1 −Φ∇2 as

dL(φ) − φ(b+ u−1B), but such a φ is exactly given by the formula

(−1)α∫

∆k

ιθ1. . . ιθk

τ2r(1⊗ exp(∧Aaff(t))×D)dt

in the case k = 1, as the proof of Proposition 4.1.8 showed.

Example 19. As an example of this construction, let us consider the Weyl algebra Wr intro-

duced in Appendix B.1. Indeed, Wr is just the universal enveloping algebra of the Lie–Rinehart

algebra (Lr,Or), where Lr = Der(Or) = Or

⟨∂/∂x1, . . . , ∂/∂xr

⟩. We denote the dual basis of L∨r

by dxi, i ∈ 1, . . . r. This Lie–Rinehart algebra admits a canonical flat Lr-connection defined

by

∇ ∂

∂xi

(∂

∂xj

)= 0, i, j = 1, . . . , r.

It is straightforward to deduce that A is given by∑

i 1⊗ ∂∂xi ⊗dxi. The Hochschild and cyclic

(co)homology ofWr are known, see B, and the image of the only nontrivial generator c2r defined

in B is, using equation B.2.2, equal to 1, the only generator of the de Rham cohomology. We

conclude that in this case the character map Φ induces an isomorphism for Hochschild and

cyclic homology.

4.2 The index theorem

4.2.1 Characteristic classes

Recall that L denotes a smooth (resp. holomorphic) Lie algebroid of rank r over a smooth (resp.

holomorphic) manifold X with structure sheaf OX and that E denotes a smooth (resp. holo-

morphic) vector bundle of rank n over X. The usual Chern–Weil construction of characteristic

classes extends to Lie algebroids in a straightforward manner if E admits a global L-connection.

When this is not the case, which is possible when L is holomorphic, the construction of charac-

teristic classes for simplicial manifolds can be applied. In our case this approach is favourable

because it is exactly the absence of a global connection that is the reason why the trace density

map maps to the Cech resolution of the Lie algebroid differential forms. Therefore, the charac-

teristic classes that we construct precisely fit to appear as the image of the trace density map

applied to the trivial element 1 as in the index theorem. We remark that smooth bundles will

be complexified in this section.

First we assume that there exists a global L-connection on E and we denote it by ∇E.

We write R∇E ∈ ΩL(End(E)) for its curvature. For any GL(K, n)-invariant polynomial Pk :



4.2. THE INDEX THEOREM 81

Mn(K) → K of degree k, the form

χ(Pk)(R(∇E)) ∈ Ω2kL

is well-defined (because of the invariance of the polynomial) and closed (because of the Bianchi

identity). The usual arguments of Chern–Weil theory apply verbatim to show that the induced

cohomology class [Pk(R(∇E))] ∈ H2k(L) is independent of the chosen connection (in fact this

follows from the simplest application of the simplicial methods). In the smooth setting this

version of Chern–Weil theory has been considered in [Cr, Fer].

The simplicial construction

Now we assume that there does not exist a global L-connection on E. Choose a cover U of X

such that E is trivializes over each Ui ∈ U , which implies that there exist local L-connections

∇i on E. Consider the pullback of the restricted Lie algebroid L|Ui0,...ik

π!L|Ui0,...,ikL|Ui0,...,ik

Ui0,...,ik × ∆k Ui0,...,ikπ

as in [MaHi]. The formula

∇totUi0,...,ik

:=

k∑i=0

ti∇i, (t0, . . . , tk) ∈ ∆k

defines a connection on the pullback π∗E|Ui0,...,ikof the restricted vector bundle E|Ui0,...,ik

.

The Chern–Weil construction for Lie algebroids is summarized in the following proposition.

Proposition 4.2.1. Let Iinv be the ring of GL(K, n)-invariant polynomials P : Mn(K) → K.

i) For P ∈ Iinv homogeneous of degree m, the forms

P(U )Ui0,...,ik:=

∫

∆k

P(R(∇totUi0,...,ik

)) ∈ Ω2m−kL |Ui0,...,ik

define a Cech cocycle in the Cech resolution C•(U ,Ω•L, dL + (−1)kδ).

ii) The cohomology class in the hypercohomology does not depend on the choice of a cover

and the choices of connections.

iii) The resulting map

Iinv → H2•Lie(L)

is a ring homomorphism.

Proof. The first claim follows from an application of Stokes’ theorem and the fact that P(R(∇totUi0,...,ik

))

is closed w.r.t. the Lie algebroid differential dπ!L:

dL

(∫

∆k

P(∇totUi0,...,ik

)

)=(−1)k

∫

∆k

dπ!L(P(∇totUi0,...,ik

) +

∫

∆k

d∆k(P(∇totUi0,...,ik

))

=

k∑i=0

(−1)j∫

∂i∆k

P(∇totUi0,...,ij,...,ik

).

To prove the second claim, we argue as in the proof of Theorem 4.1.2. Namely, given two

covers U and V , we consider the intersection of this cover; U∐

V . A diagram such as in

the proof of Theorem 4.1.2 shows that we can reduce to the case of a trivial cover with two




connections. Then the usual homotopy argument, c.f. [CF2], applies. For the last claim, we

first remark that the product on the Cech resolution of Ω•L is given by the Alexander-Whitney

product, which is graded commutative on cohomology. The Chern–Weil map factorizes through

the complex C(U ,Ωπ!L, dπ!L), c.f. [Du], and since the Chern–Weil map to this complex is a

homomorphism, the result follows.

In the following, we only need the following invariant polynomials:

Ch(X) := Tr(eX

), Td(X) := det

(x

1− ex

), A(X) = det

(x/2

sinh(x/2)

). (4.2.1)

As usual, it is meant that one takes the power series expansion around zero of the analytic

functions used in these definitions, and applies the Chern–Weil construction to each homoge-

neous component. The resulting characteristic classes are called the Chern class ChL(E) and

the Todd class TdL(E).

4.2.2 The index theorem

Some notes on the twisted case

Before we state and prove the index theorem, we make some remarks about the twisted case.

Recall that the twisted PBW map, which depends on the L-connections ∇L on L and ∇E on

E, is given by an isomorphism

jE∇ : J(L)⊗OX2E −→ SymOX

(L∨)⊗OXE.

It has the property that jE∇(φψ⊗ s) = j∇(φ)jE∇(ψ⊗ s) for φ ∈ J(L) and ψ⊗ s ∈ J(L)⊗OX2E.

The flat connection ∇(1) on J(L) extends to J(L) ⊗OX2E in a natural way. We denote the

corresponding connection on SymOX(L∨)⊗ E by D and define the difference

AE := D−∇L ⊗ 1− 1⊗∇E.

Using the property above, one can show that AE is a one form with values in

SymOX(L∨)⊗OX

EndOX(E)⊕DerOX

(SymOX(L∨)) ⊂ DiffOX

(SymOX(L∨))⊗OX

EndOX(E).

(4.2.2)

This one form can be used in exactly the same way as in the previous section to define the

twisted character map, which we will denote by ΦE. The proof that it is well-defined and does

not depend on the choices that are made applies verbatim.

Back to the theorem.

We are now in a position to state and prove the index theorem for Lie algebroids. Recall that the

character map ΦE of Theorem 4.1.2 maps -in the derived category- the sheaves of Hochschild

and cyclic complexes to the Lie algebroid complexes. Applied to the cycle 1 ∈ C0(U(L;E)), we

obtain a class

[ΦE(1)] ∈ HevLie(L)

in Lie algebroid cohomology which is canonically defined, i.e., an invariant of the Lie algebroid

L and the vector bundle E. The main theorem of this section identifies this class in terms of

the characteristic classes of the previous section:

Theorem 4.2.2. ΦE(1) = TdL(L)ChL(E).

Proof. Again we fix a cover U such that L and E are trivial when restricted to any Ui ∈ U . Let

Ui0,...,ik be an intersection. We have to compare the following two forms in Ω2n−kL |Ui0,...,ik

:

∫

∆k


)) and (−1)α∫

∆k

ιθ1. . . ιθk

τ2r(1⊗ exp(∧Aaff(t))dt.




Consider the integrand of the RHS. One can check that the evaluation morphism to Lie algebra

cohomology of Appendix B.3 commutes with the insertion operators ιθion the Hochschild

complex and the relative Lie algebra complex. This fact, and the local Riemann–Roch Theorem

B.5.1, allows us to rewrite this integrand as

ιθi1. . . ιθik

τ2r(1,A, . . . A) = ev1(ιθi1. . . ιθik

τ2r)(A, . . . , A)

= ιθi1. . . ιθik

ev1(τ2r)(A, . . . , A))

= ιθi1. . . ιθik

χ(P)(A, . . . , A),

(4.2.3)

where P is the invariant polynomial corresponding to the product of the A-genus and the Chern

class. Note that we used the stronger version of the local Riemann–Roch theorem described

in Proposition B.5.2. This is possible because of the specific form of AE that we gave in

(4.2.2). Next, in the evaluation of the Chern–Weil homomorphism, three types of curvature

terms appear:

C(A,A), C(θi, A), C(θi, θj).

As explained in B, one has C(a, b) = [π(a), π(b)] − π([a, b]) where π : Wnr → sp2r ⊕ gln is

the projection. Before we give a lemma, applicable in the non-twisted case, that allows us to

compute these terms, we fix some notation for the one form A ∈ Ω1L(DerOX

(SymOX(L∨)) that

was defined in Equation (4.1.3). We have seen that DerOX(SymOX

(L∨)) ∼= SymOX(L∨)⊗OX

L

and thus we can write

A =∑

k−1

Ak Ak ∈ Ω1L(Sym

k+1

OX(L∨)⊗ L).

Lemma 4.2.3. When the L-connection is torsion free, the lowest degree components of A are

given by

A−1 = −1⊗ idL A0 = 0,

where we view idL as an L-one-form with values in L.

Proof. We will prove this by evaluating A in an element X ∈ L, which gives a derivation of

SymOX(L∨). We apply this derivation to an element α ∈ SymOX

(L∨). Since derivations are

determined by their actions in low degrees, we can assume that α ∈ Sym1OX

(L∨). Let us recall

the PBW isomorphism in low degrees:

j∇(φ)(1) = φ(1), j∇(φ)(X) = φ(X), j∇(φ)(XY) =1

2φ(XY + YX−∇XY −∇YX).

This implies that the inverse of the PBW map satisfies

j−1∇ (α)(X) = α(X), j−1

∇ (α)(XY + YX−∇XY −∇YX) = α(XY) = 0. (4.2.4)

We use these identities to compute:

A−1(X)(α)(1) =j∇ d∇(1) j−1∇ (α)(1, X)

=X(j−1∇ (α)(1)) − j−1

∇ (α)(X)

= − α(X).

A0(X)(α)(Y) =j∇ d∇(1) j−1∇ (α)(Y, X) −∇Xα(Y)

=X(j−1∇ (α)(Y)) − j−1

∇ (α)(XY) − X(α(Y)) + α(∇XY)

=X(α(Y)) − j−1∇ (α)(XY) − X(α(Y)) + α(∇XY).

When the torsion ∇XY − ∇YX − [X, Y] vanishes Equation (4.2.4) implies that j−1∇ (α)(XY) =

α(∇XY), and hence the result follows.




In the non-twisted case, the previous lemma shows that π(A) = 0. For the twisted case, one

can show that the zeroth degree of the first component of AE in (4.2.2) can be set to zero by

changing the connection ∇E. This implies π(AE) = 0.

We proceed with the proof of Theorem 4.2.2. We assume that the L-connection is torsion

free, hence π(AE) = 0. In Lemma 2.2.10 we obtained an explicit expression for the low degrees

of θi:

θ−1i = 0, θ0i = 0, θ1i = γi.

Note that θ is not a one-form, thus γ is viewed as derivation on SymOX(L∨). The other

components of θ are derivations of higher degrees, hence π(θi) = 0. This implies that

C(A,A) = −π([A,A])

C(A, θi) = −π([A, θi])

C(θi, θj) = π([θi, θj]).

Since the commutator of derivations of degree p and q has degree p + q, it follows that

π([θi, θj]) = 0. Using the expressions from Lemma 4.1.9 we obtain:

C(A,A) = −π([A,A] = π(∇A+ R(∇aff)) = π(R(∇affL )⊗ idE + R(∇aff

E ))

C(θi, A) = −π([θi, A]) = −π(∂tiA+∇affθi + ∂ti∇aff) = −γLi − γE

i .(4.2.5)

We used here that both the operators ∇aff and ∂tj preserve the degrees of derivations. Let

us now discuss the terms in the Chern–Weil cocycle. We consider the (local) L-connections

∇ = ∇L + ∇E on the direct sum L ⊕ E. We can write the total connection in the following

form:

∇tot = dπ!L +

k∑i=0

γi.

An easy computation shows that

R(∇aff) +

k∑i=0

dtiγi.

Recall that the Chern–Weil cocycle restricted to an intersection is given by

P(U )Ui0,...,ik:=

∫

∆k


)).

Observe that the only terms in the integrand that have a contribution are the ones with exactly

k terms containing a one form dti. These terms all contribute a term γi. On the other hand,

Equation (4.2.3) shows that the class ΦE(1) is given by

ΦEUi0...ik

(1) =

∫

∆k

χ(P)(θ1, . . . , θk, A, . . . , A)dt.

The Equations in (4.2.5) show that the two integrals coincide. Finally, using the same argument

as in the proof of [W15, Prop. 20], relating the A-genus and the Todd class, the theorem is

proved.

4.2.3 Compatibility with the HKR map

In this section we shall refine the index Theorem 4.2.2 of the previous section by showing that the

Lie algebra cohomology class it determines measures the obstruction for our character map to be

compatible with the Hochschild–Kostant–Rosenberg map (4.1.1). Define HKRL := ρ∗ HKR,

the pre-composition of the HKR map with the pull-back along the anchor, i.e.,

HKRL(f0 ⊗ . . .⊗ fk) := f0dLf0 ∧ . . .∧ dLfk ∈ ΩkL, fi ∈ OX, i = 0, . . . k.

Using the obvious inclusion OX → U(L), we can compare this map with the character map of

Theorem 4.1.2:




Theorem 4.2.4. Let L be a smooth resp. holomoprhic Lie algebroid over a smooth resp.

holomorphic manifold X with structure sheaf OX. The diagram

(CC•(OX), b+ u−1B

) (Ω•

L, u−1dL

)

(CC•(U(L;E)), b+ u−1B

) (Ω2r−•

L , dL

)

HKRL

i∗ ∧TdL(L)ChL(E)

ΦE

commutes in the derived category.

Proof. We fix the usual cover U of X. We will show that the trace density map ΦEU i∗ =

TdL(L)ChL(E)|U ∧ jU HKRL where jU is the natural inclusion Ω•L → C(U ,Ω•

L). We work

over an intersection Ui0,...ik . First remark that in the definition of the character map f ∈ OX

is embedded in the sheaf of Weyl algebras W as

j∇(f) = j0∇(f) + j1∇(f) + . . . ∈ SymOX(L∨)

where j∇ is the PBW isomorphism. Analogous to the proof of Theorem 4.2.4, we now have to

consider the forms

ιθ1. . . ιθp

τ2r(A⊗ · · · ⊗A× j∇(f0)⊗ . . .⊗ j∇(fk)). ()

We will evaluate these forms using the expansion of A(∇) from Lemma 4.2.3 and the description

of θi given in Lemma 2.2.10. Write the cocycle (B.2.1) as τ2r = µ2r S2r π2r. Fix a local

basis ei, i = 1, . . . , r of L, and denote the dual basis of L∨ by dei. Let degL∨(a) and

degL(a) denote the degree in the first respectively the second component for a ∈ W(L) ∼=

SymOX(L∨)⊗SymOX

(L). For elements a0⊗. . .⊗ak we have degL(a0⊗. . .⊗ak) =∑

i degL(ai)

and degL∨(a0 ⊗ . . . ⊗ ak) =∑

i degL∨(ai). Note that µ2r is the projection on the degL =

degL∨ = 0 part of a0 ⊗ . . . ⊗ a2r followed by multiplication. We claim that we can make, in

the expression of , the following replacements;

A A−1 +A1

j∇(f0) j0∇(f0)

j1∇(fi) j1∇(fi), i 1

θl θ2l

without changing the expression. By θ2l we mean the degL∨ = 2 part of θl, which is equal to

γl. Consider Ak with k 2 in the i’th slot. We have degL(Ak) = 1 and degL∨(Ak) = k + 1,

hence an application of π2r will give an element of either (degL∨ , degL) = (k+ 1, 0) or (k, 1).

An application of esαij will only give a nonzero (0, 0) term if the term in the j’th slot has degree

(0, k) or (1, k − 1), and these terms are not present. For θl an analogous reasoning applies.

Now consider j∇(fi) in the i’th slot. The term j0∇(fi) does not survive π2r, and π2r applied to

terms jk∇ with k 2 gives terms of degree (k− 1, 0), to which an application of esαij only gives

a nonzero (0, 0) term if the term in the j’th slot has degree (0, k − 1). Again, these terms are

not present; here it is used that A0 = 0. In conclusion:

() = ιθ21. . . ιθ2

pτ2r((A−1 +A1)⊗ · · · ⊗ (A−1 +A1)× j0∇(f0)⊗ j1∇(f1)⊗ . . .⊗ j1∇(fk)).

We can still eliminate terms from this expression. We have for general elements a0⊗ . . .⊗a2r ∈W(L)⊗ . . .⊗W(L):

degL∨(αij(a0 ⊗ . . .⊗ a2r)) = degL∨(a0 ⊗ . . .⊗ a2r) − 1

degL∨(π2r(a0 ⊗ . . .⊗ a2r)) = degL∨(a0 ⊗ . . .⊗ a2r) − r




and similar formulas for the L-degree. This implies that the only terms in a0 ⊗ . . .⊗ a2r that

have a nonzero contribution after an application of τ2r have equal L and L∨ degrees. It follows

that

() = τ2r((θ2p)× . . .× (θ21)× (A1)r−k−p × (A−1)r × j0∇(f0)⊗ j1∇(f1)⊗ . . .⊗ j1∇(fk)).

The first order jets j1∇(fi) =∑

j ρ(ej)(fi)dej, i = 1, . . . , k of the fi combine with A−1 =∑i 1⊗ ei ⊗ dei, after applying π2r, to the forms ρ∗(dfi), so that we have:

() =1

k!f0ρ

∗df1 ∧ . . .∧ ρ∗dfkπ2r((θp)× . . .× (θ1)× π× . . .× π︸︷︷︸#=k

×(A−1)r−k × (A1)r−k−p)

=f0ρ∗df1 ∧ . . .∧ ρ∗dfkιθ1

. . . ιθp

ιkπk!τ2r((A−1)r−k × (A1)r−k−p)

=f0ρ∗df1 ∧ . . .∧ ρ∗dfku

−n[TdL(L)Ch(E)].

We used the definition of the cyclic cocycle using the insertion ιπ, and Theorem 4.2.2.

4.2.4 Holomorphic Lie algebroids

In this subsection we give (a sketch of) an alternative derivation of our main index theorem

using the Dolbeault-type resolution to compute the Lie algebroid cohomology constructed in

[LSX12], c.f. Example 6. We use the notations of that example, e.g. we denote the holomorphic

Lie algebroid by A. Recall the Dolbeault complex

. . . . . .

D1,0A D1,1

A . . .

D0,0A D0,1

A . . .

d1,0A

∂

d1,0A

∂

∂

d1,0A d1,0

A

∂

where Dp,qA := ∧p(A1,0)∨ ⊗ ∧q(T0,1X)∨. The kernel of the differential ∂ : D•,0

A → D•,1A is

precisely the holomorphic Lie algebroid complex (Ω•A, dA) and the total complex computes the

Lie algebroid cohomology of A.

To apply this resolution for our construction of the character map, we tensor with the

smooth jet bundle and universal enveloping algebra to obtain the complex

(J(A1,0)⊗C∞

X2U(A1,0)⊗C∞

X1D

p,qL ,∇(1) + ∂A

). (4.2.6)

This time the kernel of the differential ∂A in degree 0 is precisely the sheaf of dg algebras of

Lemma 4.1.3 for the Lie algebroid A.

Next we choose a torsion free A1,0-connection ∇ on A1,0 Remark that in this C∞-setting,

such a connection always exists. By the PBW theorem, this induces an isomorphism

j∇ : J(A1,0)∼=−→ Sym((A1,0)∨).

The non commutative version of the PBW theorem induces an isomorphism of the total space

of the dg algebra of (4.2.6) with W(A1,0)⊗C∞XD•,•

A , also denoted j∇, and allows us to introduce

A(∇) := j∇ ∇(1) j−1∇ − (∇+ ∂A).

Remark that the flat connection ∇(1)+∂A on J∞A1,0 ⊗C∞

X2U(A1,0), when transferred to WA1,0 ,

does not necessarily have the form j∇ ∇(1) j−1∇ + ∂A and hence A has both components in

the T0,1 and A1,0 direction.




Theorem 4.2.5. Let A → X be a holomorphic Lie algebroid over a complex manifold, and

E → X a holomorphic vector bundle.

i) A pair (∇A,∇E) of A-connections on A and E compatible with the holomorphic structure,

with ∇A torsion free, induces a character map

Φ∇ :(CC•(U(A;E)), b+ u−1B

)→

⊕

p+q=2r−•Dp,q(E)A[u], d

1,0A + ∂L

.

ii) For different choices of connections (∇A,∇E) and (∇A, ∇E), the maps Φ∇ and Φ∇ are

chain homotopic.

iii) (Index Theorem) The following equality of differential forms holds true:

Φ∇(1) = TdAChA(E).






Appendix A

Formality

A.1 Introduction

In this section we revisit the formality theorem for Lie algebroids as developed in a series of

papers [C, CDH, CvdB, CRvdB12, D]. The formality theorems in [CvdB, CRvdB12] are stated

and proven in great generality, namely for locally free sheaves of Lie–Rinehart algebras L of

finite constant rank over a ringed site X. The version that we will prove reads as follows.

Theorem A.1.1. Let L be a locally free sheaf of OX-modules, where OX is a fine structure

sheaf of a ringed space. There exist an L∞-quasi-isomorphism U = (U1, U2, . . .)

U : TLpoly(X)−→DL

poly(X)

with the first component equal to the HKR map. This morphism equips CL(X) with the structure

of an L∞-module over TLpoly(X), and there exists a quasi-isomorphism V = (V1, V2, . . .) of L∞-

modules

V : CL(X) −→ ΩL(X)

over TLpoly(X), with V1 equal to the HKR map for chains.

Here, U is an L∞-morphism from the dg Lie algebra of L-polyvectorfields TLpoly(X) to the dg

Lie algebra of L-polydifferential operators DLpoly(X) over X, and V is an L∞-morphism from the

dg module CL(X) of L-Hochschild chains to the dg module of L-differential forms. The first part

of this section will be a brief summary of results about (curved) L∞ algebras, their morphisms

and twists. We will prove this theorem in a more restrictive case than in [CvdB, CRvdB12],

since we don’t need it in its full generality and since our proof requires less technical machinery.

Our only application of the theorem is in the following case.

The locally ringed space (X,OX) is given by (M, SymA), where M is a smooth manifold

and A → M a smooth vector bundle. The sheaf of Lie algebroids L = DerR(Sym(A)). Local

trivializations of A and the tangent bundle TM, together with the isomorphism

DerR(Sym(A)) ∼= Γ(Sym(A)⊗ (A∨ ⊕ TM))

induced by a connection on A, c.f. §3.1.2, show that L is locally free of finite, constant rank,

and, given a TM-connection ∇A on A and a TM-connection ∇TM on TM, the following formula

defines an L-connection on L:

∇(α,X)(β, Y) = (∇AXβ,∇TM

X Y), (α,X), (β, Y) ∈ A∨ ⊕ TM. (A.1.1)

Finally, the sheaves SymA and DerR(SymA) are fine sheaves, because partitions of unity exist.

Hence, this example satisfies the conditions in the theorem.

89



90 APPENDIX A. FORMALITY

Considering the second point, the proof that we provide depends crucially on the PBW

theorem for Lie algebroids, which in turn depends on an L-connection on L. Moreover, to

globalize locally defined PBW morphisms, we use that OX and L are fine sheaves.

Remark A.1.2. Another important example is of course given by A → M a smooth Lie

algebroid, where OX = C∞(M), but this case is exactly the result from [C].

Remark A.1.3. Let us finally make some remarks about the terminology. We will be a bit

sloppy when talking about sheaves. Many of the constructions need the process of ’sheafifi-

cation’, e.g. the tensor product of sheaves is not a sheaf but a presheaf, and one defines the

tensor product of two sheaves to be the associated sheaf to this presheaf. We will usually not

mention this. Moreover, operations such as the product, contraction etc. of two given sheaves

are defined locally, i.e. over an open U ⊂ X, but again we usually won’t mention this.

A.2 Curved L∞-algebras

Let g =⊕

i gi be a Z-graded vector space. In the following we write |X| for the degree of a

homogeneous element X ∈ g. We consider the symmetric bialgebra of g, shifted by one; S(g[1]),

with the coproduct ∆ defined on generators by

∆ : X → X⊗ 1+ 1⊗ X X ∈ g[1].

Since g[1] is graded, there are signs involved in the shuffle products

∆ : X1 · · ·Xn →n∑

i=0

∑σ∈Sh(i,n−i)

±Xσ(1) · · ·Xσ(i) ⊗ Xσ(i+1) · · ·Xσ(n).

To be more precise, these signs can be reconstructed from the Koszul sign rule, which says that

for each transposition of two elements X, Y, a sign (−1)|X||Y| is introduced.

Definition A.2.1. A curved L∞-structure on g is given by a differential Q of degree 1 on

S(g[1]) which is a coderivation with respect to ∆.

Remark A.2.2. The decalage isomorphism

decn : Sn(g[1]) ∼= ∧ng[n], X1 · · ·Xn → (−1)∑n

i=1(n−i)(|Xi|−1)X1 ∧ · · ·∧ Xn

allows to define L∞-algebras in terms of the algebra Λg, but with that definition the signs are

less natural. However, when we will discuss the Taylor components, which will be introduced

hereafter, of the coderivation, morphisms of L∞-algebras, L∞-modules etc., we will use this

identification.

Since the exterior algebra S(g[1]) is cofreely generated by g, The differential Q is completely

determined by its projections

Qn : Λng → g[2− n], n 0,

called multibrackets or Taylor components of Q. These multibrackets have to satisfy identities

of the form

∑p+q=n

σ∈Sh(p,q)

±Qq+1

(Qp(Xσ(1) ∧ · · ·∧ Xσ(p))∧ Xσ(p+1) ∧ · · ·∧ Xσ(n)

)= 0,

for k 0, resulting from the condition Q2 = 0. The element Q0 ∈ g[2] is called the curvature

of the L∞-algebra, and when Q0 = 0, we simply speak of a (non curved) L∞-algebra.



A.2. CURVED L∞-ALGEBRAS 91

All the curved L∞-algebras in this work satisfy Qk = 0, k 3, and are called curved

differential graded Lie algebras (dg Lie algebras). It is customary to write the data as Q0 :=

ω ∈ g2, a degree 1 map Q1 := d : g → g and a bracket Q2 := [·, ·] : g× g → g of degree 0. The

L∞-relations above now amount to the following conditions:

[x, [y, z]] = [[x, y], z] + (−1)|x||y|[y, [x, z]] (Jacobi identity) (A.2.1a)

d([x, y]) = [dx, y] + (−1)|x|[x, dy] (derivation property) (A.2.1b)

d2 = [ω,−] (Curvature equation) (A.2.1c)

dω = 0 (Bianchi identity). (A.2.1d)

A natural example of such a curved dg Lie algebra is given by Ω•(M; End(E)), for a smooth

vector bundle E over a manifold M. The bracket is given by the wedge product of differential

forms combined with the Lie bracket on End(E). The two remaining structure maps are induced

by the choice of a connection ∇ on E: this induces a derivation on the associated bundle End(E)

and the curvature of∇ gives the element R(∇) ∈ Ω2(M; End(E)) satisfying the Bianchi identity.

Morphisms of L∞-algebras.

A morphism of curved L∞-algebras U : g1 → g2 is simply a morphism of coalgebras S(g1[1]) →S(g2[1]) compatible with differentials. Again, using cofreeness, such a morphism is determined

by its projections

Un : Λng1 → g2[1− n].

As before, compatibility with the differentials imposes conditions on the maps Un. When g1

and g2 are simply curved dg Lie algebras, these take the form U1(ω1) = ω2 and

Un+1(ω1 ∧ X1 ∧ · · ·∧ Xn) = ±dUn(X1 ∧ · · ·∧ Xn)

+

n∑i=1

±Un(X1 ∧ · · ·∧ dXi ∧ · · ·∧ Xn)

+1

2

∑p+q=n

σ∈Sh(p,q)

±[Up(Xσ(1) ∧ · · ·∧ Xσ(p)), Uq(Xσ(p+1) ∧ · · ·∧ Xσ(n))

]

+∑i<j

±Un−1([Xi, Xj]∧ · · ·∧ Xi ∧ · · ·∧ Xj ∧ · · ·∧ Xn).

(A.2.2)

If both L∞-algebras are not curved, these relations have a simple meaning in low degrees: for

n = 1 we see that U1 is a morphism of complexes. For n = 2 we find that the failure of U1 to

be compatible with Lie brackets is exact, so it induces a morphism of Lie algebras on the level

of cohomology.

L∞-modules

Let (g, Q) be a curved L∞-algebra and M a Z-graded vector space. An L∞-module structure

on M is given by a degree 1 differential ϕ which is a coderivation on the cofree Λg-module

Λg⊗M. Again we can write ϕ out in components

ϕk : Λkg⊗M → M[1− k], k 0,

and the coderivation property is given by the relations∑

p+q=nσ∈Sh(p,q)

± Xσ(1) ∧ · · ·∧ Xσ(p) ∧ϕq(Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m)

+∑

p+q=nσ∈Sh(p,q)

±(Qp(Xσ(1) ∧ · · ·∧ Xσ(p))∧ Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m)) = 0.




In degree k = 0 this means that ϕ(m) = ±ϕ0(m)±Q0 ⊗m. The fact that ϕ is a differential

leads to the following quadratic relations:∑

p+q=nσ∈Sh(p,q)

±ϕp(Xσ(1) ∧ · · ·∧ Xσ(p) ∧ϕq(Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m))

+∑

p+q=nσ∈Sh(p,q)

±ϕq+1(Qp(Xσ(1) ∧ · · ·∧ Xσ(p))∧ Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m)) = 0.

For n = 0 this means that ϕ20 = ϕ1(Q0 ⊗ −), so M is a complex when the curvature of g

vanishes.

Given two modules M and N over g, a morphism f from M to N of modules is a morphism

of comodules commuting with the coderivations. In components

fk : Λkg⊗M → N[−k]

this means that∑

p+q=nσ∈Sh(p,q)

±fq+1(Qp(Xσ(1) ∧ · · ·∧ Xσ(p))∧ Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m)

+∑

p+q=nσ∈Sh(p,q)

±fp(Xσ(1) ∧ · · ·∧ Xσ(p) ∧ϕq(Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m))

=∑

p+q=nσ∈Sh(p,q)

±ϕq(Xσ(1) ∧ · · ·∧ Xσ(p) ∧ fq(Xσ(p+1) ∧ · · ·∧ Xσ(n) ⊗m)).

Twisting by Maurer–Cartan elements

Given a curved dg Lie algebra g with structure maps Q0 = ω, Q1 = d, Q2 = [ , ], and an

element X ∈ g1, the structure maps can be twisted in the following way:

ωX = ω− dX+1

2[X,X]

dX = d+ [X, ]

[ , ]X = [ , ].

(A.2.3)

These new structure maps define another curved dg Lie algebra, as can be checked by a computa-

tion. A classical example is given by a uncurved dg Lie algebra g together with a Maurer–Cartan

element X ∈ g1 which satisfies

dX+1

2[X,X] = 0,

which means that the twisted dg Lie algebra also has zero curvature. In a curved dg Lie algebra

(g,ω, d, [ , ]) a Maurer–Cartan element is an element X ∈ g1 satisfying

dX+1

2[X,X] = ω.

In this case the twisted differerential dX := d+[X, ] has vanishing curvature: d2X = 0, therefore

the twisted dg Lie algebra (g, dX, [ , ]) is uncurved. Given an L∞-morphism U : g h of two

curved dg Lie algebras one can check that the formal sum

X :=∑k1

Uk(X, . . . , X)

k!

is an element in h1 which satisfies the Maurer–Cartan equation if X does, provided the sum

converges. In case it doesn’t converge one can throw in a formal parameter h and work in h[[ h]]

instead. Since the twisted codifferential QX is related to the originial codifferential Q by the

equation

QX = exp(−X∧) Q exp(X∧),



A.3. POLYVECTOR FIELDS AND POLYDIFFERENTIAL OPERATORS 93

we can twist the morphism U to an L∞-morphism UX : (g,ωgX, dX, [ , ]) (h,ωh

X, dX, [ , ])

defined by

UX := exp(−X∧) U exp(X∧).

Similarly, an L∞-module M over (g,ωg, d, [ , ]) is twisted to a module over (g,ωgX, dX, [ , ]) by

the codifferential

ϕX := exp(−X∧) ϕ exp(X∧),

and therefore a morphism F : M N of L∞-modules over (g, d, [ , ]) is twisted by the formula

FX := exp(−X∧) F exp(X∧)

which is a morphism of (g,ωgX, dX, [ , ])-modules.

The tangent map

Given a Maurer–Cartan element X in g, the underlying cochain complex TXg := g[1] of the

twisted dg Lie algebra with differential dX := d + [X, ] is called the tangent complex. For an

L∞-morphism U : g h the first component of the twisted morphism UX defines a tangent map

TXU : TXg → TXh of cochain complexes given explicitly by

TXU(Y) =∑k1

Uk(Y, X, . . . , X)

(k− 1)!.

A.3 Polyvector fields and polydifferential operators

We assume to be given a sheaf of Lie algebroids L over (X,OX) as described in the introduction

of this appendix, and an L-connection on L. Recall from the chapter 1 that the L-valued

polyvector fields, which form a sheaf of graded Lie algebras, are defined as

TLpoly(X) :=

⊕n−1

n+1∧OX

L,

with the zero differential and the Lie bracket given by the natural extension of the Lie bracket

on L, called the Schouten–Nijenhuis bracket. The Lie bracket has degree 0 due to the shift in

the degree. We denote this dg Lie algebra by(TLpoly(X), 0, [·, ·]S

).

Another natural sheaf of dg Lie algebras associated to a Lie algebroid L is the algebra of

L-polydifferential operators, defined by

DLpoly(X) :=

⊕−1n

U(L)⊗n+1 =⊕

−1n

U(L)⊗ . . .⊗U(L)︸︷︷︸n+1 times

,

where we use the left OX-module structure on U(L) in the definition of the tensor products.

On homogeneous elements D := D1 ⊗ . . . ⊗ Dp+1 and E := E1 ⊗ . . . ⊗ Eq+1, one defines

D i E ∈ DLpoly(X) by the formula

D i E := D1 ⊗ . . .⊗Di ⊗D(1)i+1E1 ⊗ . . .⊗D

(p+1)i+1 Eq+1 ⊗Di+2 ⊗Dp+1,

where we used Sweedler notation. With this, the Gerstenhaber bracket (of degree 0) is defined

as

[D,E]G := D E− (−1)pqE D,

where

D E :=

p∑i=0

(−1)iqD i E.




As for the ordinary shifted Hochschild complex, the Gerstenhaber bracket defines a graded Lie

algebra structure on DLpoly(X). The element B0 = 1 ⊗ 1 ∈ DL,0

poly(X) induces the product on

OX and satisfies [B0, B0]G = 0. It is easy to check that [B0, ·]G defines a differential, which is

explicitly given by

δ(D1 ⊗ . . .⊗Dn) := 1⊗D1 ⊗ . . .⊗Dn (A.3.1)

+

n∑i=1

(−1)iD1 ⊗ . . .⊗ ∆Di ⊗ . . .⊗Dn + (−1)n+1D1 ⊗ . . .⊗Dn ⊗ 1.

The triple(DL

poly(X), δ, [·, ·]G)

is a dg Lie algebra. This dg Lie algebra also admits a cup

product, which gives DLpoly(X) the structure of a Gerstenhaber algebra up to homotopy, c.f

[CRvdB12].

A.4 Differential forms and Hochschild chains

In this subsection we describe two natural dg Lie algebra-modules over the L-polyvectorfields

and L-polydifferential operators. Recall the sheaf of L-differential forms (with an unusual

grading)

ΩL(X) :=⊕n0

Λ−nOX

L∗,

which, equipped with the trivial differential and the Lie derivative of polyvectorfields, defined

by the Cartan formula:

Lγ = dL ιγ + (−1)kιγ dL, γ ∈ TL,kpoly(X),

forms a dg Lie algebra module over TLpoly(X). The complex of Hochschild L-chains is defined

as:

CL(X) :=⊕n0

J(L)⊗OX1

n.

To give this complex the structure of a dg Lie algebra module over DLpoly(X), we proceed as

in [CDH, CRvdB12]. There it is explained that the pair (DLpoly(X), CL(X)) can be realized

as the flat sections of a pair of Hochschild cochains and chains of the sheaf of algebras J(L).

The differentials and the module structure of the Hochschild (co)chains commute with the flat

connection, and thus these structures descend to flat sections. First, we consider the Hochschild

L-cochains. The second U(L)-module structure on J(L) defines an injective map

r : DLpoly(X) −→ J(L)⊗OX2

DLpoly(X)

∇(2)

∼= Dpoly(J(L)), (A.4.1)

where the RHS is given by the polydifferential operators of J(L) which are linear with respect

to the first OX-module structure. The isomorphism which is decorated by ∇(2) is given by

φ⊗D → (ψ → φD·2(ψ)), φ⊗D ∈ J(L)⊗OX2U(L), ψ ∈ J(L)

where D·2 is the action by the second U(L)-module structure on J(L). The L-connection ∇(1)

extends to Dpoly(J(L)), and the map given above induces an isomorphism of DLpoly(X) and

the flat sections of Dpoly,OX1(J(L)). For the Hochschild L-chains we proceed as follows. The

complex of Hochschild chains for the algebra J(L) is given by

CL• (X) := J(L)

⊗OX1−(•+1)

.

It is related to the L-chains by the following map:

C(J(L))ε−→ CL(X)

φ0⊗ · · · ⊗φk −→ ε(φ0)φ1⊗ · · · ⊗φk.(A.4.2)



A.5. FORMALITY FOR HOCHSCHILD L-(CO)CHAINS 95

Extending ∇(1) to C(J(L)), this map restricts to an isomorphism on the module of flat sections

on the LHS. The dg Lie algebra structure on Dpoly,OX1(J(L)) is in the same way defined as for

DLpoly(X) and it is clear that the inclusion r from (A.4.1) is compatible with the structures on

the LHS and the RHS. The standard Hochschild differential for chains is defined by

b(φ0 ⊗ · · · ⊗ φk) := φ0φ1 ⊗ · · · ⊗ φk

+

k−1∑i=1

(−1)iφ0 ⊗ · · · ⊗ φiφi+1 ⊗ · · · ⊗ φk + (−1)kφkφ0 ⊗ · · · ⊗ φk,

whereas the Dpoly,OX1(J(L))-module structure on the chain complex is defined by

LD(φ0 ⊗ · · · ⊗ φn) :=

n∑i=n−k+1

(−1)n(i+1)D(φi+1, · · · , φ0, · · · )⊗ φk+i−n ⊗ · · · ⊗ φi

+

n−k∑i=0

(−1)(k−1)(i+1)φ0 ⊗ · · · ⊗ φi ⊗D(φi+1, · · · , φi+k)⊗ φi+k+1 ⊗ · · · ⊗ φn.

Since ∇(1) acts by derivations on J(L) the differential b commutes with ∇(1), and one can

easily check that the Dpoly,OX1(J(L))-module structure L on C(J(L)) defines a DL

poly(X) mod-

ule structure on CL(X). These considerations imply that the subpair (DLpoly(X), CL(X)) of

(Dpoly(J(L)),CL• (X)) forms a dg Lie algebra together with a dg Lie algebra module, i.e we have

a commuting diagram

DLpoly(X) Dpoly(J(L))

CL(X) CL• (X)

L

r

L

s

(A.4.3)

in which the left vertical arrow is the restriction of the right one and in which s is the map to

the flat sections with respect to ∇(1) such that ε s = id, c.f. [KP]. The wavy arrows indicate

dg Lie algebra module structures.

To summarize, we described the dg Lie algebra of L-polyvector fields TLpoly(X) with the

dg Lie algebra module of L-differential forms ΩL(X) over TLpoly(X), and the dg Lie algebra of

L-polydifferential operators DLpoly(X) with the dg Lie algebra module CL

poly(X) of Hochschild

L-chains. In a diagram this looks as follows:

TLpoly(X) DL

poly(X)

ΩL(X) CL(X)

L L(A.4.4)

The next paragraph will discuss the formality theorem, which relates these two dg Lie algebras

and their modules.

A.5 Formality for Hochschild L-(co)chains

In this section we discuss the relation between the L-polyvector fields and the L-polydifferential

operators. There is an HKR type map U1 : TLpoly(X) → DL

poly(X) given by antisymmetrization

U1(X0 ∧ . . .∧ Xk) :=1

(k+ 1)!

∑σ∈Sk+1

(−1)σXσ(0) ⊗ . . .⊗ Xσ(k). (A.5.1)

This is a quasi-isomorphism of sheaves of complexes, but it is not compatible with the Lie

brackets. There is also a HKR type chain map V1 : CL(X) → ΩL(X) given by

V1(φ1 ⊗ . . .⊗ φk) := φ0dφ1 ∧ . . .∧ dφk, (A.5.2)




which is a quasi-isomorphism of sheaves of complexes. However, it is not compatible with the

dg Lie algebra-module structures.

In his seminal paper [K], Kontsevich proved that, in the case of a smooth manifold M and

L = TM, this morphism can be “corrected” to a quasi isomorphism of L∞-algebras with Taylor

components Uk : Λk(TLpoly(X)) → DL

poly(X). Amongst many other consequences, this allows to

relate the Maurer–Cartan elements in the two dg Lie algebras, which gives a correspondence

between Poisson structures and deformations of the product on C∞(M). We formulate the most

fundamental version of the so-called formality theorem, because we need it for our proof.

Local formality results

Consider the trivial Lie algebroid Rd → 0 with coordinates y1, . . . , yd. Its algebra of jets is

given by J(Rd) ∼= R[[y1, . . . , yd]] and the formal polyvectorfields Tpoly(J(Rd)) and the formal

polydifferential operators Dpoly(J(Rd)) are defined algebraically, e.g. a formal polyvector field

of degree k can be written as

X =∑

i1<...<ik

ai1...ik

∂

∂yi1

∧ · · ·∧ ∂

∂yik

, ai1...ik ∈ J(Rd).

The formality theorem [K, §7] reads as follows.

Theorem A.5.1. There exists an L∞-quasi-isomorphism

K : Tpoly(J(Rd)) −→ Dpoly(J(Rd))

with the following properties:

1. The first structure map K1 is given by the HKR map (4.1.1) .

2. K is GLd(R)-equivariant

3. Let n > 1. For any vector fields v1, . . . , vn ∈ Tpoly(J(Rd)) one has

Kn(v1, . . . , vn) = 0.

4. If n > 1, then for any vector field v ∈ T0poly(J(Rd)) linear in the coordinates yi and any

polyvector fields X2, . . . , Xn ∈ Tpoly(J(Rd)) one has

Kn(v, X2, . . . , Xn) = 0.

The formula for Kn is given by a sum over certain graphs, with weights that are given by

integrals over configuration spaces. In [K], Kontsevich already globalized the formality map to

a smooth manifold M, and discussed the compatibility of the morphism with other structures

such as the wedge product on polyvector fields.

We denote the differential forms and Hochschild chains related to the algebra J(Rd) by

ΩJ(Rd) and C(J(Rd)). Since C(J(Rd)) is a dg Lie module over Dpoly(J(Rd)), the formality mor-

phism K of Kontsevich equips C(J(Rd)) with the structure of an L∞-module over Tpoly(J(Rd).

The formality conjecture for chains, formulated by Tsygan in [Ts] and proved by Shoikhet in [S],

states that the two L∞-modules ΩJ(Rd) and C(J(Rd)) over Tpoly(J(Rd) are related as follows:

Theorem A.5.2 ([S]). There exists a quasi-isomorphism S of L∞-modules over Tpoly(J(Rd))

S : C(J(Rd)) −→ ΩJ(Rd)

with the following properties:




1. The first structure map S1 is given by the dual of the HKR map (4.1.1) .

2. S is GLd(R)-equivariant

3. If n > 1 then for any vector field v ∈ T0poly(J(Rd)) linear in the coordinates yi and any

polyvector fields X2, . . . , Xn ∈ Tpoly(J(Rd)), and any chain j ∈ C(J(Rd))

Sn(v, X2, . . . , Xn; j) = 0.

Again, the maps Sn are defined using sums over graphs with weights given by integrals

over configuration spaces. The theorem was globalized to the case of smooth manifolds, and

the compatibility between the de Rham differential and the cyclic differential B was studied by

Willwacher in [W15].

Theorem A.5.3 ([CDH, CRvdB12, C, CvdB, D]). Let (L,OX) be a locally free, fine sheaf of Lie

algebroids of constant rank r over R. There exist an L∞-quasi-isomorphism U = (U1, U2, . . .)

U : TLpoly(X)−→DL

poly(X)

with the first component equal to the HKR map U1 above. This morphism equips CL(X) with the

structure of an L∞-module over TLpoly(X), and there exists a quasi-isomorphism V = (V1, V2, . . .)

of L∞-modules

V : CL(X) −→ ΩL(X)

over TLpoly(X), with V1 equal to the HKR map V1 for chains.

Proof. The remainder of this subsection is devoted to a proof of Theorem A.5.3. It follows from

the following claim: There exist two sheaves of dg Lie algebras L1 and L2 over X, two dg Lie

modules M1, M2 over L1 and L2 respectively, and a commutative diagram

TLpoly(X) L1 L2 DL

poly(X)

ΩL(X) M1 M2 CL(X).

L L L L(A.5.3)

The hooked arrows in the upper rows are quasi-isomorphisms of dg Lie algebras and the middle

arrow in the upper row is an L∞-quasi-isomorphism. The wavy arrows indicate the dg Lie

module structures. The hooked arrows in the lower row are dg Lie module quasi-isomorphisms,

and the middle arrow in the lower row is a quasi-isomorphism of L∞-modules.

To be precise, this diagram does not provide a complete proof of the formality theorem as

stated above, since one also needs to invert the upper right and lower left arrows to obtain

the L∞-morphims. This matter is properly dealt with in [D] and [CFW], so we refer to these

articles. The proof for the existence of the diagram is divided into three steps, and we give it

because it differs slightly from existing proofs. Firstly, we stress the role played by the dual

PBW theorem for jets, and secondly, we work with curved L∞-algebras.

Step 1: Fedosov resolutions

Detailed proofs for all the statements in this step can be found in [CvdB, section 4]. Recall the

sheaf J(L) of L-jets, with its two L-module structures from chapter 2. Consider the sequence

Oα2−→ J(L)

d∇(1)−→ J(L)⊗OX2L∨

d∇(1)−→ J(L)⊗OX2Λ2L∨

d∇(1)−→ . . . (A.5.4)

where d∇(1) is the de Rham L-differential associated to the flat connection ∇(1). The filtration

on J(L) induces a filtration on the sequence, and the differential induced on the associated




graded spaces SymO(L∨) ⊗O ΛkL is the Koszul differential for SymO(L

∨). Since the Koszul

differential is exact, the sequence (A.5.4) is exact. Replacing the sheaf OX in this sequence by

TLpoly(X), D

Lpoly(X) or ΩL(X), one obtains exact sequences

TLpoly(X) TL

poly(X)⊗OX2J(L) TL

poly(X)⊗OX2J(L)⊗OX1

L∨ . . .

DLpoly(X) DL

poly(X)⊗OX2J(L) DL

poly(X)⊗OX2J(L)⊗OX1

L∨ . . .

ΩL(X) ΩL(X)⊗OX2J(L) ΩL(X)⊗OX2

J(L)⊗OX1L∨ . . .

α2d∇(1) d∇(1)

α2d∇(1) d∇(1)

α2d∇(1) d∇(1)

(A.5.5)

The map v : φ⊗ X →(ψ → φ∇(2)

X ψ)is an isomorphism of J(L)-modules

v : J(L)⊗OX2L

∼=−→ DerOX1(J(L)). (A.5.6)

Again, this is proved by studying the map on the associated graded spaces. Note that the

inclusion X → ∇(2)X is a morphism of Lie algebras. This isomorphism can be extended to

isomorphisms of dg Lie algebras:

TLpoly(X)⊗OX2

J(L) ∼= Tpoly,OX1(J(L)) =: Tpoly(J(L))

DLpoly(X)⊗OX2

J(L) ∼= Dpoly,OX1(J(L)) =: Dpoly(J(L)).

(A.5.7)

Dual to the map (A.5.6), we have the isomorphism

u : Ω1J(L) := Ω1

J(L)/OX1

∼=−→ J(L)⊗OX2Λ1L∨

which can be extended to an isomorphism of dg algebras

u :(ΩJ(L) := ΩJ(L), ddR

) ∼=−→(J(L)⊗OX2

ΛL∨,∇(2)). (A.5.8)

Recall that the module structure L of ΩL(X) over TLpoly(X) is defined as the graded commutator

of the contraction operation and the de Rham differential. The same holds for the module

structure of ΩJ(L) over Tpoly(J(L)). The inverse of the map (A.5.8) preserves the dg algebra

structure, i.e. it commutes with the de Rham differential, and we recall the following commuting

diagram [CvdB, (4.18)]

L∨ ⊗ L

O

α2

ΩJ(L)⊗DerOX1(J(L)) J(L)

where the horizontal arrows are the natural contractions, and the left vertical arrow is given

by v and u−1. The resolutions of the polyvector fields and differential forms in (A.5.5), the

isomorphisms v and u−1 and the compatibility of the dg Lie algebra module structures with

these morphisms are summarized in the following diagram:

(TLpoly(X), 0, [·, ·]

) (Tpoly(J(L))⊗OX1

ΛL∨, d∇(1) , [·, ·])

(ΩL(X), 0)(ΩJ(L) ⊗OX1

ΛL∨, d∇(1)

),

L

vα2

L

u−1α2

(A.5.9)

which is a commuting diagram of dg Lie algebras and their modules, such that the horizontal

arrows are quasi-isomorphisms.




Remark A.5.4. The brackets on the RHS are ΛL∨-linearly extended to the tensor product.

To be more precise, one has

Tpoly,ΛL∨

(J(L)⊗OX1

ΛL∨)∼= Tpoly(J(L))⊗ΛL∨

where the LHS is given by the polyvector fields of the algebra J(L)) ⊗OX1ΛL∨ which are

ΛL∨-linear, and the corresponding Schouten–Nijenhuis bracket on the RHS is given by the

ΛL∨-linear extension of the bracket on Tpoly(J(L)) to the tensor product.

To define the required structures on the Hochschild L-(co)chains we already realized them

as ∇(1)-flat sections of another dg Lie algebra with a module over it. Let us recall diagram

(A.4.3) and include the relevant operations:

(DL

poly(X), δ, [·, ·]) (

Dpoly(J(L))⊗ΛL∨, δ+ d∇(1) , [·, ·])

(CL(X), b)(CL• (X)⊗ΛL∨, b+ d∇(1)

).

L

r

L

s

(A.5.10)

Step 2: The PBW isomorphism

Recall that the PBW isomorphism is an algebra isomorphism

j∇ : J(L) −→ SymOX(L∨)

which is OX-linear with respect to the first OX-module structure on J(L) and depends on the

choice of an L-connection on L. It induces the following commuting diagram of dg Lie algebras

and their modules:

Tpoly(J(L)) Tpoly(SymOX(L∨)) Dpoly(J(L)) Dpoly(SymOX

(L∨))

ΩJ(L) ΩSymOX

(L∨)C(J(L)) C(SymOX

(L∨))

j∇

L L L

j∇

L

j∇ j∇

where the horizontal arrows are isomorphisms. The flat connection ∇(1) induces a flat connec-

tion ∇(1) = j−1∇ ∇(1) j∇ on SymOX

(L∨), and taking the tensor product of each of the objects

in the above diagram with ΛL∨, the differentials d∇(1) and d∇(1) can be included in the dg

Lie algebra (module) structures. E.g., we obtain

(Dpoly(J(L))⊗ΛL∨, δ+ d∇(1) , [·, ·]G

) j∇∼=

(Dpoly(SymOX

(L∨))⊗ΛL∨, δ+ d∇(1) , [·, ·]G).

Now set

L1 := Tpoly(SymOX(L∨))⊗OΛL∨ M1 := Ω

SymOX(L∨)

L2 := Dpoly(SymOX(L∨))⊗OΛL∨ M2 = C(SymOX

(L∨)).

The quasi-isomorphisms from (A.5.9) and (A.5.10), together with the maps induced by the

PBW map that we just discussed, combine to the hooked arrows in (A.5.3). In step 3) we will

prove the existence of the middle arrows in (A.5.3).

Step 3: Twisting the local L∞-maps

The final step of the proof is quite similar to [CDH, section 4.1], however, we include the notion

of curved L∞-algebras, which is different from their approach.




Using the fact that L is locally trivial, we can choose a trivializing basis eiri=1 of L over an

open U ⊂ X. Hence we obtain SymOX(L∨)|U ∼= O|U ⊗ Rd

formal, and similar local isomorphisms

for the polyvector fields, the polydifferential operators, the differential forms and the Hochschild

chains.

Kontsevich’ formality morphism from Theorem A.5.1 and Shoikhets formality morphisms

from Theorem A.5.2 can be used to define an O|U-linear extension of the formality morphisms

for J(Rd). By property (2) of Theorem A.5.1 and property (2) of Theorem A.5.2, these local def-

initions do not depend on the choice of basis, hence they globalize to OX-linear L∞-morphisms:

(Tpoly(SymOX

(L∨))⊗OXΛL∨, 0, [·, ·]

) (Dpoly(SymOX

(L∨))⊗OXΛL∨, δ, [·, ·]

)

(Ω

SymOX(L∨)

⊗OXΛL∨

) (C(SymOX

(L∨))⊗OXΛL∨, b

).

K

L L

S

We will twist these L∞-morphisms locally and argue that the twisted morphisms do not depend

on the local choices and thus define global L∞-morphisms.

We denote the dual basis ei of L∨ either by fi or by dei. The first notation is used

when we describe local sections of SymOX(L∨), the second is used when we denote local sections

of ΛL∨. The de Rham differential dL for L-forms with values in bundles such as SymOX(L∨)

is well-defined for L|U, although dependent on a choice of basis of L. Because K and S are

OX-linear, they commute with the de Rham differential, thus we have local L∞-morphisms

(Tpoly(SymOX

(L∨))⊗OXΛL∨, dL, [·, ·]

) (Dpoly(SymOX

(L∨))⊗OXΛL∨, dL + δ, [·, ·]

)

(Ω

SymOX(L∨)

⊗OXΛL∨, dL

) (C(SymOX

(L∨))⊗OXΛL∨, dL + b

).

K

L L

S

The horizontal arrows are quasi-isomorphisms, as can be shown by a spectral sequence argument

based on the filtration defined by the exterior degree, c.f. [CDH, section 4]. Locally we can

write the L connection on L as the sum of the de Rham differential and an endomorphism

valued one form Γ ∈ Ω1L(End(L)):

∇ = dL + Γ ijkfj ∂

∂fidek.

The element Γ can be used to twist the L∞ structures and the L∞ morphisms, as described in

(A.2.3). This gives a (local) morphism of curved L∞-algebras:

(Tpoly(SymOX

(L∨))⊗OXΛL∨,∇L, [·, ·]

) (Dpoly(SymOX

(L∨))⊗OXΛL∨, δ+∇L, [·, ·]

)

(Ω

SymOX(L∨)

⊗OXΛL∨,∇L

) (C(SymOX

(L∨))⊗OXΛL∨, b+∇L

),

K

L L

S

where the curvature term is, on both sides, given by R∇.

Lemma A.5.5. The morphisms K and S in the diagram are global L∞-quasi-isomorphims of

curved dg Lie algebras and their modules.

Proof. Property (4) from Theorem A.5.1 and property (3) from Theorem A.5.2 imply that

K = KΓ and S = SΓ , as Γ is linear in the fi-coordinates. Hence the moprhisms K and S are

globally defined morphisms of curved L∞-algebras and their modules.




For the final step we need another twist of the L∞-morphism by a Maurer–Cartan element

in the curved dg Lie algebra on the LHS. In Lemma 4.2.3 it is shown that, for a torsion free

L-connection, one has

d∇(1) = ∇− δK +∑k2

Ak

where Ak ∈ Derk−1OX

(SymOX(L∨)) and δK is the Kozsul differential for SymOX

(L∨). Recall

that a given connection can be modified to a torsion free connection. The term B := −δK +∑k2 Ak is globally defined and is clearly a Maurer–Cartan element for the curved dg Lie

algebra structure given by (∇, [·, ·]) because d∇(1) is a differential. Hence we can twist the

L∞-morphisms K and S by B. This gives global L∞-morphisms (of uncurved) dg Lie algebras

and their modules:

(Tpoly(SymOX

(L∨))⊗OXΛL∨, d∇(1) , [·, ·]

) (Dpoly(SymOX

(L∨))⊗OXΛL∨, δ+ d∇(1) , [·, ·]

)

(Ω

SymOX(L∨)

⊗OXΛL∨, d∇(1)

) (C(SymOX

(L∨))⊗OXΛL∨, b+ d∇(1)

).

KB

L L

SB

Remark that the L∞-algebra structure on the RHS is twisted by the MC-element

B :=

∞∑k=1

1

k!Kk(B, . . . , B)

where B is the Maurer–Cartan element of the LHS. In this particular case property (3) of

Theorem A.5.1 implies that B = B. From the general theory, c.f. [D], it follows that, since K

and S are quasi-isomorphisms, KB and SB are quasi isomorphims as well. Thus, KB and SB

are the middle horizontal arrows of diagram (A.5.3), which existence we showed now. Hence,

apart from inverting certain quasi-isomorphisms, Theorem A.5.3 is proved.

Step 4: Inverting the L∞-quasi isomorphisms.

Since L∞-quasi isomorphisms are invertible up to homotopy, the upper right arrow and the lower

left arrow in diagram (A.5.3) can be inverted to obtain the L∞-morphisms U and V from the

theorem. However, this is not completely straightforward, and for a discussion of the subtleties

we refer to [CFW, D]. In [D] an explicit inverse is constructed via a recursive procedure. This

explicit choice of an inverse allows Dolgushev to prove that the formality map thus obtained

is invariant under an action of a smooth Lie group, provided the connection is invariant under

this action. We will use this property in our main application to prove that the formality map

preserves the polynomial degrees , c.f . §3.2.

Remark A.5.6. We would like to make a short comparison with the techniques which are based

on formal geometry that Fedosov, and later for example [C] and [D] used. In that approach,

one starts with the observation that

TLpoly(X) −→

(TLpoly(X)⊗ SymOX

(L∨)⊗ΛL∨,−δK

)(A.5.11)

is a resolution, where −δK is the fiberwise Koszul differential of SymOX(L∨). The isomorphism

TLpoly(X) ⊗ SymOX

(L∨) ∼= Tpoly,OX(SymOX

(L∨), equips the RHS with a fiberwise Lie algebra

structure, but the inclusion in (A.5.11) is not a morphism of Lie algebras. Given a torsion free

L-connection on L, the differential −δfiber +∇ can be, using an iterative procedure, adjusted

to a differential

D = −δfiber +∇+A




where A ∈ Ω1L(DerOX

(SymOX(L))). Moreover, the inclusion in (A.5.11) can be modified to a

resolution

TLpoly(X) −→

(TLpoly(X)⊗ SymOX

(L∗)⊗ΛL∗, D)

which respects the Lie algebra structures. The PBW isomorphism allows for more direct con-

struction of D, which in our case is just given by d∇(1) . The constructions are carried out in a

similar fashion for DLpoly(X) etc., and Fedosov applied it to the Weyl algebra bundle associated

to a symplectic manifold.

A.6 Cyclic L-chains

Finally, let us discuss the extension to cyclic chains. It is a straightforward extension, mostly

based on [W11]. It starts with the observation that the complex C(J(L)) carries an additional

cyclic differential B turning it into a mixed complex: we just take the standard B-operator

defined by

B(φ0 ⊗ · · · ⊗ φk) :=

n∑j=0

(−1)nj1⊗ φj ⊗ · · · ⊗ φn ⊗ φ0 ⊗ · · · ⊗ φj−1

with the remark that φ−1 := φn. Note that we use the augemented complex given by

C(J(L))−kaug := J(L)⊗OX1

(J(L)/OX1)k, which gives the same homology as the complex C(J(L)),

but has a simpler expression for the cyclic differential B. This operator commutes with the

Grothendieck connection, i.e.,

[∇(1)X , B] = 0, for all X ∈ L

since ∇(1)X (1) = 0. Thus it defines an operator, which we also denote by B, on CL(X). Recall

from 1.2.2 that cyclic complexes are defined as CL(X)⊗K[u−1]W, equipped with the differential

b+ u−1B where W is a K[u−1]-module.

Proposition A.6.1. The u−1-linear extension of V induces a morphism

V :(CL(L)[[u−1]]⊗C[u−1] W, b+ u−1B

)→

(Ω•

L[[u−1]]⊗C[u−1] W, u−1d

)

of L∞-modules over TLpoly(X).

Proof. The proof is very similar to the proof of [W11, cor. 4]; we give a sketch. We need to

show that the lower arrows of diagram (A.5.3);

ΩL(X)W MW

1 MW2 CL(X)

WSB

(A.6.1)

intertwine u−1dL onΩWL = ΩL[[u

−1]]⊗K[u−1]W, the OX-linear de Rham differential u−1ddR on

MW1 = Ω

SymOX(L∨)

⊗K[u−1]W, the OX-linear cyclic differential u−1B onMW

2 = C(SymOX(L∨))⊗K[u−1]

W, and u−1B on CL(X)⊗K[u−1] W respectively.

• The morphism ΩL(X)W → MW

1 is the composition of the inclusion u−1 α2 : ΩL(X) →ΩJ(L) and the PBW map j∇. It is easy to check that u−1 α2 -note that u−1 indicates a

morphism, and not a formal variable, intertwines the differentials dL and ddR, and since

j∇ is an algebra isomorphism it intertwines ddR for J(L) and ddR for SymOX(L∨).

• The main theorem in [W11] states that the middle arrow commutes with the respective

differentials.

• Finally, the morphism CL(X)W → MW

2 is given by the composition j∇ s. The map s

intertwines the differentials by definition, and j∇ commutes with the differentials because

it is an algebra isomorphism.



Appendix B

The local Riemann–Roch

theorem

In this appendix we describe, following [PoPfTa10, W15] the fundamental cyclic cocycle on the

Weyl algebra and recall the Riemann–Roch theorem for this cocycle.

B.1 The Weyl algebra

Let Or = K[x1, . . . , xr] be the polynomial algebra on r generators. The Weyl algebra Wr

is, by definition, the algebra of differential operators on Or. Over Or, it is generated by the

fundamental derivations ∂/∂xi, i = 1, . . . , r subject to the well-known commutation relations

[∂

∂xi, xj] = δij.

It therefore admits another presentation as the space of polynomials K[q1, . . . , qr, p1, . . . , pr]

equipped with the Moyal–Weyl product

f g := m eπ(f⊗ g), f, g ∈ K[q1, . . . , qr, p1, . . . , pr], (B.1.1)

where m is the commutative product of polynomials and

π :=

r∑i=1

(∂

∂pi⊗ ∂

∂qi−

∂

∂qi⊗ ∂

∂pi

).

The isomorphism with Wr is simply given by sending xi → qi, pi → ∂/∂xi.

B.2 The Hochschild cocycle

Using the spectral sequence associated to the filtration given by the degree of a polynomial, it

was proved in [FT] that

HH•(Wr) =

K • = 2r

0 • = 2r.

A generator for the nontrivial class in degree 2n is given by

c2r :=∑

σ∈S2r

(−1)σ1⊗ yσ(1) ⊗ . . . yσ(2r), y2i = qi, y2i−1 = pi, i = 1, . . . , r.

103



104 APPENDIX B. THE LOCAL RIEMANN–ROCH THEOREM

In [FFS], a dual Hochschild cocycle generating the only nontrivial class was defined by the

formula

τHoch2r (a) := µ2r

∫

∆2r

∏0i<j2r

e(tj−ti−12)πji(1⊗ π∧r)(a)dt1 · · ·dt2r, (B.2.1)

where a = a0 ⊗ . . . a2r ∈ W⊗(2r+1)r and µ2r(a0 ⊗ . . . a2r) = a0(0) · · ·a2r(0) is the evaluation

at zero followed by the commutative multiplication. That this cocycle exists follows simply by

duality, and indeed we have the property that

τ2r(c2r) = 1. (B.2.2)

However, the explicit expression above allows to prove the surprising fact the cocycle τHoch2r is

gl(r,K)-invariant and basic:

LXτHoch2r = 0, τHoch

2r (. . . , X, . . .) = 0, X ∈ gl(r,K). (B.2.3)

The center of the Weyl algebra is clearly equal to K, and, since every derivation is inner, as

an explicit calculation shows, we have the exact sequence of Lie algebras

0 −→ K −→ Wrad−→ Der(Wr) −→ 0. (B.2.4)

B.3 The map to Lie algebra cohomology

Let g be a Lie algebra over K. The Chevalley–Eilenberg cochain complex, written C•Lie(g), is

a special case of the complex of L-differential forms (1.1.2) in the case L = g and (X,OX) =

(pt,K). Let C be a unital, associative algebra over K. Consider the cyclic cochain complex

CC•W(C) for a K[u]–module W. The evaluation map

ev1(φ)(a1, . . . , ak) :=∑σ∈Sk

(−1)σφ(1⊗ aσ(1) ⊗ . . .⊗ aσ(k)), a1, . . . ak ∈ C,

defines a morphism of cochain complexes

ev1: (CC•W(C), b+ uB) −→

(C•

Lie(gl(C))[[u]]⊗K[u] W,dLie

), (B.3.1)

where gl(C) denotes the Lie algebra associated to C, i.e., C equipped with the commutator.

B.4 The fundamental cyclic cocycle

In [PoPfTa10] and [W15] the Hochschild cocycle τHoch2r was extended to a full cyclic cocycle in

the (b, B)-complex. Here we follow the presentation of [W15]. Define the insertion operator

ιπ =

r∑i=1

ιpiιqi : Ck(Wr,W

∗r ) → Ck−2(Wr,W

∗r ).

With this notation, for each choice of w ∈ W\0, the cochain

τw2r := e−uιπτHoch2r ⊗w ∈ CC2r

W(Wr)

is closed: (b+ uB)τw2r = 0. For w = 1 in ordinary cyclic cohomology one can expand

τ12r = τHoch2r + uτ2r−2 + . . .+ urτ0 ∈ C2r(Wr)[u]

with τ2r−2k = ιkπτHoch2r /k!. It is possible to perform the integration to obtain an explicit

formula for for the lower degrees τ2k as an integral over the simplex ∆2k analogous to (B.2.1),

c.f. [PoPfTa10].



B.5. THE LOCAL RIEMANN–ROCH THEOREM 105

For the twisted case, we need a slight generalization of this cocyle. The algebra Wnr :=

Mn(Wr) is Morita equivalent to Wr, thus has the same Hochschild and cyclic homology. The

cyclic cocyle can be generalized by the formula

τw,n((a0 ⊗M0)⊗ . . .⊗ (ak ⊗Mk)) := τw2r(a0 ⊗ . . .⊗ ak)tr(M0 . . .Mk).

The algebra Wnr contains the Lie subalgebra sp2r ⊕ gln, and the cocycle is basic and invariant

w.r.t. this subalgebra, c.f. [W15].

B.5 The local Riemann–Roch theorem

The most fundamental property of the cyclic cocycle τw2r is the local Riemann–Roch theorem

which links it in Lie algebra cohomology to the characteristic classes appearing in the index

theorem. This theorem has a long history dating back to [FT, Thm 5.1.1.], where it first

appeared in abstract form for a Hochschild class of τHoch2r . Other appearances of the theorem

are in [BNT, FFS, PoPfTa10, W15]. We shall state the theorem below as an equality on the

level of chains (the last proposition), the proof follows as in [W15] by evaluating the integrals

appearing in the formula for τw2r.

Let us first recall the Chern–Weil homomorphism in Lie algebra cohomology: Let h ⊂ g be

an inclusion of Lie algebras, and suppose that there exists an h-equivariant projection π : g → h.

Then π determines, in the language of Lie algebroids, a g-connection on h by the formula

∇XZ = π([X,Z]), for X ∈ g, Z ∈ h,

with curvature R ∈ Hom(Λ2g, h) given by

C(X, Y) = [π(X), π(Y)] − π([X, Y]), X, Y ∈ g.

The Chern–Weil construction then leads to a homomorphism

χ :(Sym• h∨

)h → C2•Lie(g, h),

where the RHS is the relative Chevalley–Eilenberg complex. Explicitly, it is given by the formula

χ(P)(X1, . . . , X2k) :=1

k!

∑σ∈S2k

σ(2i−1)<σ(2i)

(−1)σP(C(Xσ(1), Xσ(2)), . . . , C(Xσ(2k−1), Xσ(2k))

),

where X1, . . . , X2k ∈ g and P ∈ Symk h∨. For the Riemann–Roch theorem we need this con-

struction for the inclusion

glr ⊕ gln ⊂ Wnr

where glr is embedded as diagonal matrices (by the inclusion glr → sp2r) and gln as constant

matrices. Recall the invariant polynomials defining the Chern and Todd classes:

Ch(X) := Tr(eX

), Td(X) := det

(x

1− eX

), A(X) =

x/2

sinh x/2(B.5.1)

As usual, it is meant that one takes the power series expansion around zero of the analytic func-

tions used in these definitions, and applies the Chern–Weil construction to each homogeneous

component.

Theorem B.5.1 (Local Riemann–Roch). Under the u-linear extension of the morphism (B.3.1),

we have the following equality of Lie algebra classes:

[ev1(τn)]2k = (−1)k[χ(Ar Chn)]2k ∈ H2k

Lie (gl(Wnr )⊕ gln(Or); glr ⊕ gln) .



106 APPENDIX B. THE LOCAL RIEMANN–ROCH THEOREM

This theorem is proved by the following proposition, which is in fact stronger than the

theorem itself. Let Cn,r be the subalgebra of Wnr consisting of polynomials of the form∑r

j=1 fj(q)⊗ 1+∑

j1 gj(q)⊗Mj for fj and gj polynomials in q and Mj ∈ gln.

Proposition B.5.2 ([FFS, W15]). On the subalgebra Cn,r the cocycles ev1(τn)2k and χ((Ar Chn)2k)

agree.



Bibliography

[AC] C. A. Abad and M. Crainic, Representations of Lie algebroids up to homotopy, J. Reine

Angew. Math. 663, pp. 91—126 (2012).

[BB] A. Beilinson and J. Bernstein, A proof of Jantzen conjectures, Adv. in Sov. Mat. 16, pp.

1–50 (1993).

[BP] A. Blom, H. Posthuma, An index theorem for Lie algebroids,

http://arxiv.org/pdf/1512.07863.pdf.

[BNT] P. Bressler, R. Nest and B. Tsygan, Riemann–Roch theorems via Deformation Quanti-

zation, I & II., Adv. Math. 67, pp. 1–25 & 26–73 (2002).

[Br] J.L Brylinski, A differential complex for Poisson manifolds, J. Diff. Geom. 28, no. 1, pp.

93–114 (1988).

[BrGe] J. L. Brylinski, and E. Getzler, The homology of algebras of pseudodifferential symbols

and the noncommutative residue, K–Theory 1, pp. 385–403 (1987).

[C] D. Calaque, Formality for Lie algebroids, Comm. Math. Phys. 3, pp. 563–578 (2005).

[CvdB] D. Calaque and M. van den Bergh, Hochschild cohomology and Atiyah classes, Adv.

Math. 224, no. 5, pp. 1839–1889 (2010).

[CRvdB10] D. Calaque, C.A. Rossi and M. van den Bergh, Hochschild (co)homology for Lie

algebroids, Int. Math. Res. Not., no. 21 2010, , pp. 4098–4136.

[CRvdB12] D. Calaque, C.A. Rossi and M. van den Bergh, Caldararu’s conjecture and Tsygan’s

formality, Annals of Math. 176 (2012), no. 2, 865-923.

[CDH] D. Calaque, V. Dolgushev and G. Halbout, Formality theorems for Hochschild chains

in the Lie algebroid setting, J. Reine Angew. Math. 612, pp. 81–127 (2007).

[CdSW] A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras,

Berkeley Math. Lect. Notes, V. 10 (1999).

[CFW] A. Cattaneo, G. Felder and T. Willwacher, The character map in deformation quatiza-

tion, Adv. Math. 228, no. 4, pp. 1966–1989 (2011).

[CoMo] A. Connes and M. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic

groups. Topology 29, no. 3 (1990).

[Cr] M. Crainic. Differentiable and algebroid cohomology, van Est isomorphisms, and charac-

teristic classes, Comment. Math. Helv. 78, no. 4, pp. 681–721 (2003).

[CF1] M. Crainic and R. Fernandes, Integrability of Lie brackets, Annals of Math. 157, no. 2,

pp. 575–620 (2003).

107



108 BIBLIOGRAPHY

[CF2] M. Crainic and R. Fernandes, Exotic Characteristic Classes of Lie Algebroids, Lecture

Notes in Phys., 662, pp. 157–176 (2005).

[CF3] M. Crainic and R. Fernandes, Integrability of Poisson brackets, J. Diff. Geom. 66, pp.

71–137 (2004).

[CM] M. Crainic and I. Moerdijk, Deformations of Lie brackets: cohomological aspects, J. Eur.

Math. Soc. 10, pp. 1037–1059 (2008).

[Co] A. Connes, Non commutative differential geometry Inst. Hautes Etudes Sci. Publ. Math.

62, pp. 41–144, (1985).

[D] V. A. Dolgushev, A formality theorem for Hochschild Chains, Adv. Math. 200, no. 1, pp.

51–101 (2006).

[DTTs] V. A. Dolgushev, D. Tamarkin and B. Tsygan, Formality theorems for Hochschild

complexes and their applications, Lett. Math. Phys. 90, pp. 103—136 (2009).

[Du] J.L. Dupont. Simplicial de Rham cohomology and characteristic classes of flat bundles,

Topology 15 (1976) pp. 233–245

[EnFe] M. Engeli, G. Felder, A Riemann–Roch-Hirzebruch formula for traces of differential

operators., Ann. Sci. Ec. Norm. Super. 4, pp. 621–653 (2008).

[ELW] S. Evens, J. H. Lu, and A. Weinstein, Transverse measures, the modular class and a

cohomology pairing for Lie algebroids, Quart. J. Math. Oxford 50, no. 2, pp. 417–436 (1999).

[Fed] B. Fedosov, A simple geometrical construction of deformation quantization, J. Diff. Geom.

40, no. 2, pp. 213–238 (1994).

[FFS] B. Feigin, G. Felder, and B. Shoikhet, Hochschild cohomology of the Weyl algebra and

traces in deformation quantization, Duke Math. J. 127, no. 3, pp. 487–517 (2005).

[FT] Feigin, B. and B. Tsygan: Riemann-Roch theorem and Lie algebra cohomology, Rend.

Circ. Math. Palermo 2 Suppl. No. 21, pp. 15–52 (1989).

[Fer] R. Fernandes. Lie algebroids, holonomy and characteristic classes, Adv. Math. 170, no.

1, (2002).

[GJ] E. Getzler and J.D.S. Jones, A∞-algebras and the cyclic bar complex, Illinois J. Math. 34,

pp. 256–283 (1989).

[HKR] G. Hochschild, B. Kostant and A. Rosenberg, Differential forms on regular affine alge-

bras, Trans. Amer. Math. Soc. 102, pp. 383-–408 (1962)

[Hue] J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin- Vilkovisky

algebras, Ann. de l’Institut Fourier 48, pp. 425—440 (1998).

[Ka] C. Kassel, L’homologie cyclique des algebres enveloppantes, Invent. Math. 91, pp. 221–251

(1988).

[K] M. Kontsevich, Deformation quantisation of Poisson manifolds, Lett. Math. Phys. 66, no.

3, pp. 157–216 (2003).

[KK] N. Kowalzig and U. Krahmer, Duality and products in algebraic (co)homology theories.,

J. of Alg. 323, no. 7, pp. 2063–2081 (2010)

[KP] N. Kowalzig and H. Posthuma, The cyclic theory of Hopf algebroids, J. Noncomm. Geom.

5, pp. 423–476 (2011).



BIBLIOGRAPHY 109

[LSX08] C. Laurent–Gengoux, M. Stienon and P. Xu, Holomorphic Poisson Manifolds and

Holomorphic Lie Algebroids, Int. Math. Res. Not. 2008, pp. 1-–46 (2008).

[LSX12] C. Laurent–Gengoux, M. Stienon and P. Xu, Exponential map and L∞ algebra associ-

ated to a Lie pair, C. R. Math. Acad. Sci. Paris 350, no. 17–18, pp. 817–821 (2012).

[LSX14] C. Laurent–Gengoux, M. Stienon and P. Xu, Poincare–Birkhoff–Witt isomorphisms

and Kapranov dg-manifolds, arXiv:1408.2903 (2014).

[LiS] H.-Y. Liao and M. Stienon, Emmrich–Weinstein theorem for graded manifolds,

arXiv:1508.02780 (2015).

[LiSX] H.-Y. Liao, M. Stienon, and P. Xu, Kontsevich–Duflo theorem for Lie pairs,

arXiv:1605.09722 (2016).

[L-BM] D. Li-Bland and E. Meinrencken, On the van Est homomorphism for Lie groupoids,

arxiv.org/abs/1403.1191v1 (2014).

[L] A. Lichnerowicz, Les varietals de Poisson et leurs algebres de Lie associees J. Diff. Geom.

12, pp. 253—300 (1977).

[LoQ] J.-L. Loday and D. L. Quillen, Cyclic homology and the Lie algebra homology of matrices,

Comment. Math. Helv. 59, pp. 569–591 (1984).

[Lo] J. Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften Vol. 301,

Springer, (1998).

[M] K.C.H. Mackenzie, General theory of Lie groupoids and Lie algebroids London Math. Soc.

Lect. Not. Ser. 213, Cambridge University Press, (2005).

[MaHi] K. Mackenzie and P. Higgins. Algebraic constructions in the category of Lie Algebroids,

J. of Alg. 129, pp. 194–230 (1990).

[Mar] C.-M. Marle, Calculus on Lie algebroids, Lie groupoids and Poisson manifolds, Diss.

Math. 457, Inst. Math., Polish Acad. Sci., (2008).

[Me] R. Melrose, The Atiyah–Patodi–Singer Index Theorem, Res. Not. in Math. 4, A K Peters,

Ltd., Wellesley, (1993).

[MoMr] I. Moerdijk and J. Mrcun, Introduction to Foliations and Lie Groupoids, Cam. Stud.

in Adv. Math. 91, Cambridge University Press, (2003).

[NeTs95] R. Nest and B. Tsygan, Algebraic index theorem, Comm. Math. Phys. 172, pp. 223—

262 (1995). [

[NeTs96] R. Nest and B. Tsygan, Formal versus analytic index theorems, Intern. Math. Res.

Not. 11, pp. 557–564 (1996).

[NeTs01] R. Nest and B. Tsygan, Deformations of symplectic Lie algebroids, deformations of

holomorphic symplectic structures, and index theorems. Asian J. Math. 5, no. 4, pp. 599–635

(2001).

[NeWa] N. Neumaier and S. Waldmann, Deformation quantization of Poisson structures asso-

ciated to Lie algebroids, Symm. Int. Geom. Methods Appl. (SIGMA) 5 (2009).

[NWX] V. Nistor, A. Weinstein and P. Xu, Pseudodifferential operators on differential

groupoids, Pacific J. Math. 189, pp. 117–152 (1999).



110 BIBLIOGRAPHY

[PoPfTa10] M. Pflaum, H. Posthuma and X. Tang, Cyclic cocycles on deformation quantizations

and higher index theorems, Adv. Math. 6, pp. 1958–2021 (2010).

[PoPfTa15] M. Pflaum, H. Posthuma and X. Tang, The localized longitudinal index theorem for

Lie groupoids and the van Est map Adv. in Math. 270, pp. 223—262 (2015).

[Pr] J. Pradines, Theorie de Lie pour les groupoıdes differentiables. Relations entre proprietes

locales et globales, C. R. Acad. Sci. Paris Ser. A-B 263, pp. A907–A910 (1966).

[P] B. Pym, Poisson structures and Lie algebroids in complex geometry PhD thesis, University

of Toronto (2013)

[Qu] D. Quillen, Superconnections and the Chern character, Topology 24, pp. 89—95 (1985).

[Rin] G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math.

Soc. 108 pp. 195–222 (1963).

[Se] J.-P. Serre, Lie algebras and Lie groups Lect. Not. in Math. 1500 Springer–Verlag (1992).

[S] B. Shoikhet, A proof of the Tsygan formality conjecture for chains. Adv. Math. 179 (2003)

no 1. 7-37

[SX] M. Stienon and P. Xu, Fedosov dg–manifolds associated with a Lie pair, arXiv:1605.09656

[math. QA] (2016).

[Tr] F. Treves, Topological vector spaces, distributions, and kernels, Dover Publications (2006).

[Ts] B. Tsygan, Formality conjectures for chains, in Differential topology, infinite-dimensional

Lie algebras, and applications Amer. Math. Soc. 194, pp. 261–274 (1999).

[We] C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math-

ematics 38, Cambridge Univ. Press (1995).

[W15] T. Willwacher, Cyclic cohomology of the Weyl algebra, J. of Alg. 425, pp. 277–312

(2015).

[W11] T. Willwacher, Formality of Cyclic Chains, Int. Math. Res. Notices 17, pp. 3939–3956

(2011).

[Wo] M. Wodzicki. Cyclic homology of differential operators, Duke Math. J. 2, pp. 641–647

(1987).

[Xu99] P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys.

200:3, pp. 545–560 (1999).



Samenvatting

Cyklische theorie van Lie algebroıden

Dit proefschrift richt zich op de studie van de cyklische theorie van de universeel omhullende

algebras van Lie algebroıden. Lie algebroıden zijn meetkundige objecten die infinitesimale

symmetrieen beschrijven, en het concept omvat een groot aantal klassieke begrippen uit de

meetkunde, zoals Poisson varieteiten, foliaties en acties van Lie algebras op varieteiten. Het

bestuderen van meetkundige objecten is in veel gevallen equivalent aan het bestuderen van de

algebras van functies op deze objecten, en deze observatie heeft geleid tot de niet-commutatieve

meetkunde, waarbij men niet-commutatieve algebras, die niet noodzakelijk zijn gerelateerd aan

een meetkundig object, bestudeert met technieken die voortkomen uit de meetkunde. Voor

elke Lie algebroıde kan men een niet-commutatieve algebra -de universeel omhullende algebra-

definieren, welke onder andere de algebra van differentiaaloperatoren op een varieteit en de

universeel omhullende algebra van een Lie algebra veralgemeniseert. In dit proefschrift laten

we zien dat de cyklische theorie van deze algebra gelijk is aan de Poisson (co)homologie van de

duale van de Lie algebroıde, welke op haar beurt gelijk is aan de Lie algebroıde cohomologie

met waarden in de symmetrische algebra van de geadjungeerde representatie tot op homotopy

(gedraaid door een lijnbundel). Verder definieren we een spoor-dichtheid afbeelding van de

cyklische theorie van de universeel omhullende algebra naar het de Rham complex van de Lie

algebroıde, wat een veralgemenisering is van bekende resultaten voor de raakbundel van een

variteit. We gebruiken een Cech resolutie van het de Rham complex, wat de constructie ook

toepasbaar maakt op holomorfe Lie algebroıden. Zowel de berekening van de cyklische theorie

van de universeel onhullende algebra als de constructie van de spoor-dichtheid afbeelding is

gebaseerd op de Poincare–Birkhoff–Witt stelling voor Lie algebroıden, die we dan ook eerst

bewijzen.

111



Summary


In this thesis we study the cyclic theory of universal enveloping algebras of Lie algebroids.

Lie algebroids are geometrical objects that encode infinitesimal symmetries, and the concept

encompasses many classical objects from geometry, such as Poisson manifolds, foliations and

actions of Lie algebras on manifolds. The study of geometrical objects is in many cases equiv-

alent to the study of the algebras of functions on these objects, and this observation led to the

field of noncommutative geometry, where one studies noncommutative algebras, that are not

necessarily related to geometrical objects, with techniques from geometry. For each Lie alge-

broid on can define a noncommutative algebra, called the universal enveloping algebra, which

generalizes the algebra of differential operators on a manifold and the universal enveloping al-

gebra of a Lie algebra. In this thesis we show that the cyclic theory of this algebra is equal

to the Poisson (co)homology of the dual of the Lie algebroid, which in turn is equal to the Lie

algebroid cohomology with values in the symmetric algebra of the adjoint representation up to

homotopy (twisted by a line bundle). Moreover, we define a trace-density map from the cyclic

theory of the universal enveloping algebra to the de Rham complex of the Lie algebroid, which

generalizes known results for the tangent bundle of a manifold. We use a Cech resolution of

the de Rham complex, which makes the construction suitable for holomorphic Lie algebroids

as well. Both the calculation of the cyclic theory of the universal enveloping algebra as well as

the construction of the trace-density map is based on the Poincare–Birkhoff–Witt theorem for

Lie algebroids, which we therefore prove first.

112

508195-L-os-Blom508195-L-os-Blom508195-L-os-Blom508195-L-os-Blom Processed on: 20-3-2017Processed on: 20-3-2017Processed on: 20-3-2017Processed on: 20-3-2017

Cyclic theory of Lie algebroids Arie Blom


Arie B

lom

Boekenlegger nog aan te leveren door klant

508195-os-Blom.indd 1,6 10-02-17 11:25

Documents

UvA-DARE (Digital Academic Repository) Cyclic theory of ... · 508195-L-bw-Blom CYCLIC THEORY OF LIE ALGEBROIDS ... Prof. dr. S. V. Shadrin ... Allereerst wil ik Hessel Posthuma bijzonder